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A wavelet approach to cardiac signal
processing for low-power
hardware applications
Joël M.H. Karel

©Copyright Joël M.H. Karel, Maastricht 2009
Universitaire Pers Maastricht
ISBN 978-90-5278-887-6
All rights reserved. No part of this thesis may be reproduced, stored in a retrieval system of
any nature, or transmitted in any form by any means, electronic, mechanical, photocopying,
recording or otherwise, included a complete or partial transcription, without the permission
of the author.

A WAVELET APPROACH TO CARDIAC SIGNAL PROCESSING FOR
A WAVELETLOW-POWER APPROACH TO HARDWARE CARDIAC APPLICATIONS
SIGNAL PROCESSING FOR
LOW-POWER HARDWARE APPLICATIONS
PROEFSCHRIFT
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Universiteit Maastricht, op
ter gezag verkrijging van de van Rector de graad Magnificus, van doctor Prof. aan mr. deG.P.M.F. Universiteit Mols Maastricht, volgens het op
besluit gezag van de hetRector CollegeMagnificus, van Decanen, Prof. inmr. hetG.P.M.F. openbaarMols te verdedigen volgens het op
besluit van hetdinsdag College15 vandecember Decanen, 2009 in het omopenbaar 14.00 uur te verdedigen op
dinsdag 15 december 2009 om 14.00 uur
door
door
Joël Matheus Hendrikus Karel
Joël Matheus Hendrikus Karel
UUNIVERSITAIRE
PERS MAASTRICHT
P
M

Promotor:
Prof. dr. ir. R.L.M. Peeters
Copromotor:
Dr. R.L. Westra
Beoordelingscommissie:
Prof. dr. H. Kingma (voorzitter)
Prof. dr. M.P.F. Berger
Prof. dr. Y. Rudy (Washington University in St. Louis)
Dr. ir. W.A. Serdijn (Technische Universiteit Delft)
Dr. P.G.A. Volders
The research reported in this thesis has been funded by Technology Foundation STW (project
number DTC 6418)

Contents
Contents
i
1 Introduction 3
1.1 Research setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Research questions and thesis outline . . . . . . . . . . . . . . . . . . . . . 4
2 Cardiac signal processing 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The human heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Electrocardiogram data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 The Electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Signal archives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Cardiac rhythms and pathologies . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Laplace and z-transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Filtering signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Wavelet transformations 25
3.1 Continuous wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Wavelets from filter banks . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Perfect reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.2 Orthogonal filter banks . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.3 Multi-resolution analysis . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.4 Wavelet and scaling functions . . . . . . . . . . . . . . . . . . . . . 31
3.2.5 Vanishing moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.6 Linear phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.7 Polyphase filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The stationary wavelet transform . . . . . . . . . . . . . . . . . . . . . . . 36
i

ii
CONTENTS
3.4 Multiwavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Analog implementation of wavelets 43
4.1 Dynamic translinear systems . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Wavelet transformations as linear systems . . . . . . . . . . . . . . . . . . 45
4.3 Padé approximation of wavelet functions . . . . . . . . . . . . . . . . . . . 48
4.4 L2 approximation of wavelet functions . . . . . . . . . . . . . . . . . . . . 51
4.4.1 Wavelets and the L2 space . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.2 Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.3 Vanishing moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4.4 Obtaining a good starting point . . . . . . . . . . . . . . . . . . . 57
4.5 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Orthogonal wavelet design 71
5.1 Measures for the quality of a given representation . . . . . . . . . . . . . . 72
5.2 Wavelet parameterization and design . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.2 Enforcing additional vanishing moments . . . . . . . . . . . . . . . 79
5.2.3 Design and optimization . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.4 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Multiwavelet parameterization and design . . . . . . . . . . . . . . . . . . 89
5.3.1 Parameterization of lossless systems . . . . . . . . . . . . . . . . . 90
5.3.2 Parameterization of scalar wavelets . . . . . . . . . . . . . . . . . . 95
5.3.3 Parameterization of multiwavelets . . . . . . . . . . . . . . . . . . 98
5.3.4 Balanced vanishing moments . . . . . . . . . . . . . . . . . . . . . 101
5.3.5 Multiwavelet design . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Biomedical applications of wavelet design 109
6.1 Applications of wavelet design in cardiology . . . . . . . . . . . . . . . . . 109
6.1.1 Detecting the QRS complex using orthogonal wavelet design . . . 109
6.1.2 QT time measurement using designed multiwavelets . . . . . . . . 113
6.2 Bias field removal from MR images using wavelet design . . . . . . . . . . 118
6.2.1 Radio Frequency inhomogeneities in magnetic resonance images . . 118
6.2.2 Bias field removal in magnetic resonance images . . . . . . . . . . 118
6.2.3 Wavelet design for RF inhomogeneity detection . . . . . . . . . . . 119
6.2.4 Filtering MR images with designed wavelets . . . . . . . . . . . . . 121
7 Conclusions and directions for further research 125
Bibliography 129
Summary 139
Samenvatting 141

CONTENTS
iii
Curriculum Vitae 143
Lists of Symbols and Abbreviations 145
Index 149

Acknowledgements
The BioSens project is a cooperation with the Maastricht University and the Delft
University of Technology. It is financially supported by the Technology Foundation
STW (project number DTC 6418), applied science division of NWO and the technology
programme of the Dutch Ministry of Economic Affairs. In addition it is supported
by the following research partners: Medtronic Bakken Research Center Maastricht,
Medtronic Subcutaneous Diagnostics & Monitoring in Arnhem, Maastricht Instruments
B.V., Twente Medical Systems International B.V. in Enschede, SystematIC design B.V.
in Delft and Weijand R&D Consultancy, B.V. in Hellevoetsluis.
A Ph.D. thesis is a long-term project and it is impossible to complete it without the
support of many people around you. I would like to express my gratitude to all of them.
My gratitude goes to my promotor and friend Ralf Peeters who was an incredible
source of new ideas and made the link between the design of orthogonal wavelets and
lossless systems in which he is an expert. My colleague Jordi Heijman was a great help
regarding the background on cardiology.
Furthermore I would like to express my gratitude to Ronald Westra, my direct colleague
and the project leader of the BioSens project in Maastricht, who, together with
Wouter Serdijn and Richard Houben made the BioSens project possible. As a copromotor
Ronald enthusiastically proofread several sections of my thesis and helped to make
them more enjoyable to read.
A special word of thanks goes to my colleagues from the Delft University of Technology:
Sandro Haddad and Wouter Serdijn. Our kind cooperation lead to a considerable
number of publications. The trips to Delft were always enjoyable.
To the industry partners. In particular Richard Houben, Jan Peuscher, Frans Smeets
and Kiwi Smit who’s active participation in the users committee contributed to the
success of this project.
To my colleagues of the Department of Knowledge Engineering Who supported me
throughout my Ph.D. time and ensured that I had enough time beside the educational
load to complete my thesis.
My gratitude also goes to Orazio Gambino who provided the data and supported me
with the application on bias field correction.
1

2 CONTENTS
And last but not least to my friends, family, parents and wife. They endured a long
period in which I had little time for them, but also ensured that I was able to take my
distance from my thesis when necessary. Due to unforseen circumstances, handing in the
full draft of my Ph.D. thesis to my promotor and sending the manuscript to the reading
committee almost coincided with the date of our marriage and the birth of our daughters
Romy and Sofia, respectively. However my wife Erica courageously withstood the extra
stress involved.
Joël Karel

Chapter 1
Introduction
Paraphrasing Jane Austen [9], one can state that: It is a truth universally acknowledged,
that people die. Moreover, it is an established fact that among the causes of death, the
cardiovascular related diseases worldwide rank as number one [123]. Technical advances,
such as pacemakers, help to reduce the share of these diseases in the total number of
deaths. Paramount in the future advancement of pacemakers and similar implantable
devices, is the progress that can be achieved in their ability to sift, process and interpret
sensor data. In this light, this work addresses two relevant aspects of new generation
implantable devices: low-power implementation combined with high computing power,
and efficient signal processing, using wavelets and various aspects of systems theory.
1.1 Research setting
For readers unfamiliar with cardiology a brief introduction is provided in Sections 2.2–
2.4. One possibility of interest for decreasing the mortality among patients at risk of
a cardiovascular disorder, is the implantation of a therapeutic device or a monitoring
device. A classical example of such a therapeutic device is a pacemaker [122, 51], which
is further discussed in Section 2.1. The fact that permanent artificial pacemakers are implanted
in the human body makes it currently still impossible to recharge them, limiting
the lifespan of the device to the battery lifetime.
Nowadays a sensing circuit is present that controls the frequency at which a therapy is
applied, contributing to the patients’ comfort. The signal processing techniques employed
in production pacemakers are relatively simple [51] in general. More advanced algorithms
have the clear potential to give a considerable increase in patient comfort by tuning the
mode of operation to the electrophysiological state of the heart. However, since the
required sensing circuit is always active [49], this introduces a new source of power
consumption. The power consumption tends to increase with the complexity of the sense
amplifier, and power saving techniques are therefore a relevant topic.
3

4 CHAPTER 1. INTRODUCTION
In the last few decades a novel signal processing technique called wavelet transforms
has emerged. As is well known, the classical Fourier transform (see Section 2.6) represents
a signal in terms of its frequency components. Wavelets, as discussed in Chapter 3, on
the other hand represent the signal in terms of both pseudo-frequency (scale) and place
(time). In addition, wavelet analysis is capable of flexibly adapting its focus, such that
it narrows the time window and widens the frequency window for high frequencies, and
widens the time window and narrows the frequency window for low frequencies. This
is referred to as the “zoom-in” property of wavelets. Wavelets are nowadays widely
used in the field of cardiac signal processing [61, 78, 83, 102, 98, 112, 87] and signal
processing in general. However, for a realistic integration in implantable devices, a powerefficient
implementation is essential. As argued in Chapter 4, it is opportunistic, from
a power consumption perspective, to perform as many computations as possible in the
analog domain. This is due to the fact that analog-to-digital converters account for a
considerable proportion of the power consumption.
1.2 Research questions and thesis outline
Three main research questions are addressed in this thesis:
1. How to approximate wavelets for the implementation in continuous-time analog
systems.
In earlier work [53, 50, 49, 54, 48] analog dynamic translinear systems (see Section
4.1) were used as a platform to implement continuous wavelet transforms (see
Section 3.1). This involved the approximation of the wavelet transform with the impulse
response of a linear system as discussed in Section 4.2. A brief background on
linear systems is provided in Section 2.8. Using the technique of Padé approximation
the authors of [53] obtained a rational approximation of the Laplace transform
of a given wavelet function. In Section 4.3 this approach, that is only suited for the
approximation of a limited number of wavelet functions, is discussed in more detail.
A new approach, based on L 2 approximation, that can be used for a wider range of
wavelet functions is discussed in Sections 4.4 and 4.5. This approach can even be
employed for the approximation of the wavelet function of discrete wavelets such
as the Daubechies 3 wavelet as demonstrated in Section 4.5. The approximation of
discrete wavelets is possible due to the fact that the dilation and wavelet equation,
that are discussed in Section 3.2, provide a connection between discrete-time and
continuous-time. This is illustrated in Figure 1.1
2. How to design (multi)wavelets for an application at hand.
Another important issue of interest is the choice of a wavelet basis [103]. When employing
the Fourier transform the basis functions are fixed, i.e., sines and cosines.
However for wavelets this is not the case and the basis needs to satisfy the conditions
associated with the wavelet framework. One possibility for wavelet selection
is to use a parameterized wavelet that can be tuned to the application at hand

1.2. RESEARCH QUESTIONS AND THESIS OUTLINE 5
Continuous
wavelets
Discrete
wavelets
Wavelet function
Dilation and wavelet eqn
Orthogonal Filter
bank
L2 approximation
Linear system
approximation of
wavelet
transform
Circuit design
Design
Low-power,
analog DTL
implementation
Wavelet
parameterization
Prototype signal
(application dependent)
Applications
Design criterion
Sparsity
Weighting
Figure 1.1: Overview of the design and analog implementation of wavelets. Continuoustime
wavelets can be approximated by matching the associated wavelet function with the
impulse response of a linear system. This can then be implemented using the dynamic
translinear circuit approach. Orthogonal discrete wavelets have an associated filter bank.
Under conditions, using the dilation and wavelet equations, an associated wavelet function
can be found that can be approximated. For orthogonal wavelets from filter banks a
parameterization and a design criterion are available that allow for the design of wavelets.

6 CHAPTER 1. INTRODUCTION
[101]. Another possibility is to design a custom wavelet as is the case in this study
[69]. Earlier work on the design of wavelets matched the amplitude spectrum and
the phase spectrum of a wavelet to a reference signal in the Fourier domain separately
[24]. The authors of [46] describe an algorithm for the design of biorthogonal
and semi-orthogonal wavelets. The design criterion comes down to the maximization
of the energy in the approximation coefficients, however no clear motivation
is provided by the authors. In [92] a parameterization of orthogonal wavelets, involving
polyphase filters and the lattice structure as for example in [107] (see also
Section 3.2), was discussed, however no clear design criterion was provided. In Section
5.1 two design criteria are introduced for discrete wavelets, that can be used
to measure the quality of a discrete wavelet for representing a given signal. In this
context, a wavelet of “good quality” ensures that the signal’s energy is clustered
in the wavelet (time-frequency) domain, i.e., it maximizes sparsity. In Section 5.2
a parameterization of discrete-time orthogonal wavelets involving polyphase filterbanks
and the lattice structure as in [92, 107] is discussed. The enforcement of
additional vanishing moments and the use of the design criterion is discussed as
well. An integral approach for orthogonal wavelet filter design is developed and
presented in that section. As discussed in [68] and illustrated in Figure 1.1 it is
possible to compute a wavelet function associated with the designed wavelet filter
bank. If this wavelet function is sufficiently smooth, it is possible to approximate
the designed wavelet in analog circuits.
When multiple well-defined features in a signal need to be distinguished simultaneously,
multiwavelets [76, 77, 107] can be employed. In the multiwavelet multiple
waveforms are used to jointly generate a basis for the L 2 space. In Section 3.4
it is discussed how these multiwavelets can be parameterized in polyphase terms
and how the condition of orthogonality comes down to requiring that a system in
polyphase is “lossless” [117]. In order to design multiwavelets, a parameterization of
lossless systems is required, as is discussed in Section 5.3. For this parameterization
in Section 5.3.1 the “tangential Schur algorithm” [55] is used. This parameterization
is first provided for scalar wavelets in Section 5.3.2 and for multiwavelets in
Section 5.3.3. One of the issues involved is the enforcement of a first vanishing
moment which is a requirement for a valid wavelet multiresolution structure.
3. To investigate the practical potential of this approach for cardiac applications.
In Section 6.1 two practical applications of (multi)wavelet design with respect to
cardiac signal processing are discussed. In Section 6.1.1 it is demonstrated how
scalar wavelet design is useful for the detection of QRS complexes in ECG signals.
In Section 6.1.2 the length of the QT interval is estimated using designed
multiwavelets. In Section 6.2 the use of wavelet design for an application in 2D
image analysis is discussed. This describes the removal of an artifact in magnetic
resonance imaging, called the “bias field”, using specially designed wavelets [64].

Chapter 2
Cardiac signal processing
In the first part of this chapter a brief introduction is provided for readers unfamiliar
with cardiology in Sections 2.2–2.4. One possibility of interest to decrease the mortality
of patients at risk of a cardiovascular disorder, is the implantation of a therapeutic device
or a monitoring device. A classical example of such a therapeutic device is a pacemaker
[122, 51], as is further discussed in Section 2.1. For this research various datasets with
ECG data were used as discussed in Section 2.3. In Section 2.4 it is discussed that due
to pathologies various morphologies and rhythms may be exhibited in ECG signals.
In the second part a short introduction to signal processing is provided in Section 2.5.
Three classical transforms of interest in this work are the Fourier transform (Section 2.6),
the Laplace transform (Section 2.7) and the z-transform (Section 2.7). A brief background
in linear systems is given in Section 2.8, followed by an introduction to filters in
Section 2.9.
2.1 Introduction
Cardiovascular related diseases are still the major cause of death in The Netherlands,
as well as worldwide, according to WHO data [123], accounting for 33% of the total
mortalities in 2004 [60]. On average this means over 45,000 death per year or over a
hundred deaths per day in The Netherlands alone. However the proportion of deaths due
to cardiovascular disorders is decreasing. Prevention, better health care and technological
advances account for this development.
Among these technological advances are new drugs, therapies, monitoring devices and
implantable therapeutic and diagnostic devices. These devices measure signals that may
give an indication of the state of operation of the heart. However, since these signals
are often corrupted with noise, a denoising step may be required. Denoising is one of
the aspects addressed by the field of Signal processing and, in general, monitoring and
diagnostic devices rely heavily on signal processing techniques. For diagnostic purposes
7

8 CHAPTER 2. CARDIAC SIGNAL PROCESSING
it may be necessary to store the signals that have been measured. Some devices, such
as implantable diagnostic and monitoring devices, may detect events in order to decide
whether or not to store the signals that are being measured.
The pacemaker is the most prominent implantable therapeutic device. Annually over
600,000 pacemakers are implanted worldwide [122]. Early artificial pacemakers did not
possess a sensing circuit and only consisted of a power source, a pulse generator and an
electrode [51]. They simply delivered a fixed-rate pulse to the heart, regardless of the state
of the heart. Since this type of pacemaker can compete with the spontaneous activity of
the heart, it could cause arrhythmias and other undesired effects. In addition the therapy
was also applied during normal operation of the heart, i.e., when it was unnecessary. A
consequence of this, is that the built-in battery was depleted relatively fast. To overcome
these shortcomings, demand pacemakers were developed that do take intrinsic heart
activity into account (see e.g. [51]). As a result of this development, nowadays a sensing
circuit is present in pacemakers. In this sensing circuit signal processing is performed as
part of the decision stage to determine whether or not stimuli are needed. Therapeutic
devices may be critical in order to sustain the life of the patient, therefore their decision
algorithms have to be very robust. Failure of these devices may result in lawsuits and
costly claims. As a result medical companies are conservative if it comes to employing
new technologies [51] in implantable devices.
Because a persons life depends on the correct interpretation of the signals by the
pacemaker and other medical devices, it can be concluded that signal processing is of
great importance to the field of cardiology.
2.2 The human heart
The human heart is a special organ. It has a size that is slightly bigger than a fist and it
mainly consists of muscle tissue, the myocardium, that contracts up to several billions of
times during a persons lifespan. The muscle tissue of which the heart consists, is unique
in the human body. It is a special kind of involuntary muscle. The heart basically acts
as a pump that consists of four chambers. Oxygen-poor blood enters the heart in the
right atrium (RA). This blood is pumped to the right ventricle (RV). From there it is
pumped through the lungs for oxygenation and this oxygenated blood reenters the heart
in the left atrium (LA). From there it is pumped to the left ventricle (LV), which pumps
the oxygenated blood through the rest of the body. Since the atria only pump the blood
to the adjacent ventricles their tissue mass is relatively small. In addition the tissue
mass of the RV is small compared to that of the LV since the RV only has to pump the
blood through the lungs whereas the LV has to pump the blood through the whole body.
The cells of the myocardium conduct current and if stimulated by an electrical pulse
these muscle cells contract. This is called excitation-contraction coupling [14]. In their
resting state these cells are said to be polarized and their electrical potential is at resting
potential. If electrical activation of a cell is initiated, Na + and Ca + ions travel through
ion channels in a membrane in and out the cell. This process is called depolarization.

2.2. THE HUMAN HEART 9
aortic
valve
SA node
pulmonary
valve
AV node
RA
RV
LA
LV
atrioventricular
valves
Interventricular septum
+ His bundle
Figure 2.1: Schematic representation of the heart
In the case of ventricular depolarization the Na + ions accounts for 99% of the effect.
During the 150-300 ms after the heart cell is depolarized, the dynamic balance of the
electrical currents maintains what is known as an action potential. This action potential
ends with the repolarization of the cell, which involves moving ions into and out of the
cell in the opposite direction, allowing the cell to conduct electrical impulses again. For
a more detailed background on electrophysiology the reader is referred to [124].
In order for the heart to effectively pump the blood through the body, the contractions
of the muscle cells have to be coordinated. Special pacemaker cells control the heart rate.
This process goes through a number of steps during normal activity:
i) The sino-atrial (SA) node, located at the top of the RA generates an electrical
impulse that propagates relatively slowly through the muscles of the atria, such
that these are depolarized. The conduction of this pulse from the AV node to the
LA tracts mainly over Bachmann’s bundle.
ii) When this electrical pulse reaches the atrio-ventricular (AV) node after traversing
from the SA node over the anterior, middle, and posterior internodal tracts, the

10 CHAPTER 2. CARDIAC SIGNAL PROCESSING
propagation is delayed. During this delay the transfer of blood from the atria to
the ventricles is completed.
iii) The pulse travels through the His-bundle, the bundle branch and the Purkinje
fibers to the endocardium of the ventricles.
iv) The ventricles contract from bottom to top stimulated by a fast moving pulse.
The anisotropy of the tissue is effective here; the tissue of the heart has a certain
orientation and the electrical pulses travel faster along this direction.
If the regular coordination in the cardiac activity is disturbed this is called an arrythmia.
The excitation and conduction system consists of the SA node, the internodal tracks,
Bachmann’s bundle, the AV node, the bundle of His, bundle branches and Purkinje fibers.
The AV node is innervated by the autonomic nervous system, however the intrinsic rate
of the SA node during normal sinus rhythm is the highest and as a result under nonpathological
circumstances the SA node controls the heart rate. Certain chemicals, e.g.
epinephrine, may affect the heart rate.
2.3 Electrocardiogram data
A number of different measurements can be conducted on the cardiovascular system.
The electrical activity of the heart can be measured with the electrocardiogram (ECG or
EKG). The heart sounds or vibrations can be measured as the phonocardiogram, and
the pressure waves propagating through the arteries as the arterial pulse. The ECG will
be discussed in this section, where the reader is referred to [100] for a more thorough
discussion on this subject.
2.3.1 The Electrocardiogram
A signal of particular interest in this research is the ECG. The ECG is the measurement
of the electrical potential between various points on or in the body. The measurement
electrodes are called leads. There are various possible set-ups for lead placement. Typically
a 12-lead ECG is acquired by placing the leads on the body (surface ECG). Since
the heart is a 3-dimensional (3D) object it is important that the ECG provides information
about the hearts activity in three (approximately) orthogonal directions [36]. The
standard lead placement for the 12-lead ECG ensures that this spatial information is
provided. For a more thorough discussion on lead placement the reader is referred to for
example [100, 35]. An example of the 12-lead ECG plus three extra leads for the PTB
database (see Section 2.3.2) is displayed in Figure 2.2.
It is also possible to measure the ECG by placing an electrode directly on the tissue
of the heart. One then speaks of the intracardiac ECG (IECG) [51]. The measurement
of an IECG is obviously more invasive than the measurement of the surface ECG. Yet
another possibility for placing leads is to implant a monitoring device under the skin that
records the subcutaneous ECG.

i
2.3. ELECTROCARDIOGRAM DATA 11
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1000 2000 3000 4000
1000 2000 3000 4000
Figure 2.2: Excerpt of the ECG of patient 1 from the PTB database over 4500 ms. The
conventional 12 lead ECG is displayed (leads i, ii, iii, avr, avl, avf, v1, v2, v3, v4, v5, v6)
along the three Frank leads (vx, vy, vz). The ECG was digitized at 1 kHz. In this ECG,
right-bundle branch block is visible.

12 CHAPTER 2. CARDIAC SIGNAL PROCESSING
QT
ST
P
PQ
QRS
T
U
Figure 2.3: Segments of the ECG
From the ECG recordings various waves can be distinguished that correspond to
the propagation of electrical activity across the heart as described in Section 2.2. The
morphology and variability of these waves can vary from lead to lead due to the fact that
the heart is a 3D organ. Also the muscle mass is not uniform throughout the heart, and as
a consequence it is easier to see the activity of the ventricles than of the atria. The leads
may also pick up signals from other sources, such as respiration, muscle movement and
electrode movement. Respiration can cause low-frequency noise. But muscle movement,
and to a lesser extent electrode movement, may cause in-band noise which can be much
harder to filter out. The heart may also suffer from an arrythmia that gives rise to a
different morphology. A number of examples of morphologically different ECG waves is
displayed in Figure 2.4.
Due to the step-wise operation of the heart during normal sinus rhythm, a number
of physiological meaningful segments can be identified in the ECG (see for example [59])
as displayed in Figure 2.3:
P wave Depolarization of the atria. Corresponds to step i on page 10.
PQ segment Propagation delay in the AV node (step ii) on page 10) corresponds to
iso-electric segment.
QRS complex Ventricular depolarization (steps iii and iv on page 10). Due to the
muscle mass of the ventricles this generally is the dominant complex. Meanwhile
atrial repolarization occurs.
ST segment Ventricular muscle cells maintain their action potential resulting in an
iso-electric segment.
T wave Ventricular repolarization.
Sometimes other small amplitude waves such as the “U wave” are visible [59].
A number of intervals of interest can be derived from these segments:

2.3. ELECTROCARDIOGRAM DATA 13
1200
1100
1000
1200
1100
1000
1100
1050
1000
950
50 100 150 200 250
50 100 150 200 250
50 100 150 200 250
1300
1200
1100
1000
1200
1100
1000
1100
1000
900
800
700
50 100 150 200 250
50 100 150 200 250
50 100 150 200 250
1200
1100
1000
1100
1000
900
1200
1100
1000
900
50 100 150 200 250
50 100 150 200 250
50 100 150 200 250
Figure 2.4: Examples of smoothed ECG beats from the MIT-BIH arrythmia database.
QT interval Duration of ventricular depolarization and repolarization. Also indicated
as the duration of the ventricular electrical systole, which is the electrical activity
that stimulates the ventricles to contract. During the diastole the myocardium
relaxes.
RR interval Duration of a complete ventricular cardiac cycle.
PP interval Duration of a complete atrial cardiac cycle.
Due to the physical meaning of these time segments their proper identification and
duration is essential for the assessment of the patients condition. In addition the various
segments may have a certain morphology. The duration of the segments and their
morphology may be related to a certain rhythm; that may in turn correspond to a certain
arrythmia. In clinical situations a cardiologist judges the ECG, and using his/her
expert knowledge, determines the rhythm. However this decision is a challenging task
for automated systems. In particular if the requirement is that the system has to work
in real-time, and especially if it concerns an implantable device.

14 CHAPTER 2. CARDIAC SIGNAL PROCESSING
2.3.2 Signal archives
The main source of data used in this study are the Physionet signal archives [43]. This
is a collection of databases of various types of signals, of which ECG recordings form the
majority. The most prominent recordings are the MIT-BIH databases that were recorded
by the Beth Israel Deaconess Medical Center and Massachusetts Institute of Technology.
In particular the MIT-BIH arrythmia database is widely used for both the evaluation of
arrhythmia detectors as well as for basic research into cardiac dynamics. Additionally
the Physikalisch-Technische Bundesanstalt (PTB) database [16] which is also available
through Physionet has been used in this research. This same database was used for
the Computers in Cardiology 2006 Challenge [88] on QT interval estimation. One of
the participants of this challenge developed a set of manual annotations that has been
published in [25].
In addition datasets from industry partners were available. Medtronic supplied a
number of datasets that were recorded with a Medtronic RevealR subcutaneous leadless
ECG recorder.
2.4 Cardiac rhythms and pathologies
From the morphology and frequency of the various complexes in the ECG, cardiologists
can determine the involved cardiac rhythm. Furthermore the patient may suffer
from a certain pathology that is associated with the rhythm at hand, or with a specific
morphology of one of the complexes in the ECG. In order to arrive at more effective therapies
the diagnostic capabilities of therapeutic devices have to be improved. Detecting
morphologies in ECGs may be of great importance for expanding these capabilities.
The human heart is a syncytium, which means that electrical pulses can propagate
freely from cell to cell in any direction on the myocardium. In normal sinus rhythm this
propagation is done in an orderly fashion. However if this process is disturbed, turbulent
and disorganized propagation of the waves causes the myocardium to contract in a chaotic
manner; it fibrillates. This can occur in the atria, hence atrial fibrillation (AF). Often
AF is asymptomatic, i.e., the patient does not experience any grief from this arrythmia.
However, there is an elevated risk of stroke since due to the poor circulation of blood
during AF the blood may pool and clot. Ventricular fibrillation (VF), puts the patient
in acute risk of sudden cardiac death. In order to restore normal sinus rhythm in case
of VF, a defibrillator is used which generates a large electrical discharge to “reset” the
heart. For patients at chronic risk of VF, a cardioverter-defibrillator can be implanted
that monitors the heart, and in case an arrhythmia is detected, therapy is applied.
There are a large number of arrhythmias and pathologies that can occur [13]. For
effective diagnosis and treatment, signal processing is a very important tool, certainly in
the absence of a physician when automated diagnosis is required such as in implantable
medical devices.

2.5. SIGNAL PROCESSING 15
2.5 Signal processing
Signal processing is the field that aims to analyze, manipulate and interpret signals.
This includes the removal of noise, separation of signals from various sources, feature
extraction, storage (optionally with compression) and signal representation.
Signals can be either continuous-time or discrete-time, with either analog or digital
values. It is popular convention not to make the distinction between whether the
signal is sampled (continuous/discrete-time) and how these samples are quantified (analog/digital).
Instead “analog” is a popular term for continuous-time analog signals and
“digital” is a popular term for discrete-time digital signals. In this work the terms “analog”
and “digital” will refer to the representation of the signal values, regardless whether
they are sampled or not.
Most sensor information concerns continuous-time analog signals with the signal values
represented in voltage or in ampères. In order to obtain a digital signal, the sensor
data has to be quantified. This is done by an analog-to-digital (A/D) converter. This
converter has a certain number of bits that together with the range of both the converter
and the signal determines the precision it can obtain.
The A/D converter usually also samples the input signal such that the output signal
becomes a discrete-time digital signal. The number of samples per time unit is the sampling
rate or the sampling frequency. The Nyquist frequency is a property of a discretetime
system and defined as half the sampling frequency. The Nyquist rate [94, 104] is twice
the highest frequency that can be perfectly reconstructed in a continuous-time signal,
using interpolation, from a discrete-time signal which has been sampled at a frequency
equal to this Nyquist rate. So if the Nyquist frequency (half of the sampling frequency)
of the sampled system is larger than the bandwidth (half of the Nyquist rate) of the
continuous-time source signal, then perfect reconstruction should, at least in theory, be
possible.
Signals can be represented in various domains. Sensor information such as ECGs is
usually represented in the time domain. If one wants to examine certain characteristics of
the signal it can be opportunistic to examine the signal in a different domain. For example
in order to view the frequency content of a signal one can use the frequency domain, or
if one wants to analyze time information and frequency information simultaneously, one
can use the wavelet domain. To convert a signal to the frequency domain or to the
wavelet domain, the Fourier (or Laplace) transform or the wavelet transform can be
used, respectively.
2.6 Fourier transforms
The Fourier transform is a linear operator that maps a finite energy time domain function
f(t) to a frequency domain function F (ω) [15, 17]. The function f(t) is expressed in terms
of F (ω) as:
f(t) = √ 1 ∫ ∞
F (ω)e iωt dω, (2.1)
2π
−∞

16 CHAPTER 2. CARDIAC SIGNAL PROCESSING
where ω is the angular frequency variable (in radians per second) and i is the complex
number. This formula is used in the “synthesis”.
The Fourier transform (used in the “analysis”) is defined as:
F (ω) = 1 √
2π
∫ ∞
−∞
e −iωt f(t)dt, (2.2)
Note that the product of the normalization factors of (2.2) and (2.1) has to be equal to
1
2π
. In this case the normalization factors are chosen to be symmetrical.
A time domain signal f(t) is said to have finite energy if its L 2 -norm is finite:
√ ∫ ∞
||f|| 2 = |f(t)| 2 dt < ∞ (2.3)
−∞
The angular frequency ω is related to the frequency h in Hertz (Hz) by the identities
ω = 2πh and h = ω 2π . (2.4)
When F (ω) is expressed in polar coordinates as F (ω) = r(ω)e iφ(ω) , further meaning
can be given to the complex function F (ω) (2.1). The function r(ω) represents the
magnitude, i.e. the amplitude, of the complex harmonics e iωt per frequency, and φ(ω)
their initial phase angles. This can be expressed in a pair of Bode plots. The Bode
magnitude plot shows the logarithm of the magnitude in relation to the frequency, plotted
on a logarithmic frequency axis. The frequency response of a linear, time-invariant system
(see Section 2.8) can for example be visualized in such manner. The Bode phase plot
shows the initial phase angle in relation to the frequency, again plotted on a logarithmic
frequency axis.
Using Euler’s formula (see e.g. [39])
e iφ = cos φ + i sin φ, (2.5)
it becomes clear that the Fourier transform decomposes the signal in terms of (orthogonal)
sine and cosine basis functions.
If the Fourier transformed signal has compact support in the sense that there exists
an angular frequency ω 0 such that:
F (ω) = 0, ∀|ω| > ω 0 , (2.6)
then the signal is said to be bandlimited with bandwidth ω 0 .
If the function f is periodic then it will obviously have infinite energy. However the
power, i.e., the energy per time unit, can still be required to be finite. The Fourier
transform over a single period will then be calculated and one then sometimes speaks of
a Fourier series. However it is popular convention to reserve this term for the discrete
Fourier transform of a periodic function.

2.6. FOURIER TRANSFORMS 17
Of the many interesting properties of the Fourier transform, one of particular interest
is that convolution in the time domain of two functions f and g
h(t) = (f ∗ g)(t) =
∫ ∞
−∞
becomes multiplication in the Fourier domain:
f(τ)g(t − τ)dτ, (2.7)
H(ω) = √ 2πF (ω)G(ω), (2.8)
where F (ω), G(ω) and H(ω) are the Fourier transforms of f(t), g(t) and h(t) respectively.
This property is particularly useful in the field of signal processing where convolution
is extensively used. The Fourier transform has many more properties. For more details
the reader is referred to for example [15, 84].
When determining the discrete Fourier transform (analysis) X of a (possibly complex)
discrete-time signal x = {x[0], . . . , x[N − 1]}, one treats this finite length signal x as
if it were periodic. If the complex harmonics e −iΩn that are used as a basis for the
decomposition are chosen such that an integer number of periods fit in the length of the
signal x, this indeed gives the desired result. One then speaks of the discrete Fourier
series. The discrete Fourier series is defined as:
X[k] = √ 1
N−1
∑
x[n]e −iΩn , (2.9)
N
where Ω = 2πk/N, k = 0, 0, . . . , N − 1. Its inverse transform (synthesis) is given by:
n=0
x[n] = √ 1
N−1
∑
X[k]e iΩn . (2.10)
N
k=0
For a pair of vectors (e.g. finite length discrete-time signals) a and b the convolution
product c = a ∗ b is defined as the vector:
c[n] = (a ∗ b)[n] =
n∑
a[l]b[n − l], n ∈ {0, 1, . . . , N − 1}. (2.11)
l=0
In the Fourier domain this again becomes multiplication:
C[k] = √ NA[k]B[k], (2.12)
where A, B and C are the Fourier series of a, b and c respectively.
Since the basis functions of the Fourier transform are sinusoids that extend over
the entire real line R, the Fourier transform is not well equipped for studying transient
phenomena. A classical way around this issue is to use a windowing function w(t) to
arrive at:
F w (τ, ω) = √ 1 ∫ ∞
w(t − τ)e −iωt f(t)dt, (2.13)
2π
−∞

18 CHAPTER 2. CARDIAC SIGNAL PROCESSING
This transform is called the windowed Fourier transform alias the short time Fourier
transform [40]. The time window w(t − τ) localizes the transform around time τ and the
exponential e −iωt in frequency around frequency ω. Due to the Heisenberg uncertainty
principle [58] from the field of quantum dynamics, one cannot obtain an arbitrary accurate
time-frequency measurement. Instead the product of the “standard deviations”
of the time- and frequency windows, i.e., the area of the uncertainty rectangle, obeys a
certain lower bound: σ τ σ ω ≥ 1 2
. This area is for instance minimal if w(t) is a Gaussian
function, i.e., in the case of the Gabor transform. For the windowed Fourier transform
the uncertainty rectangle has constant dimensions and a variable placement in the timefrequency
plane (see for example [80]).
2.7 Laplace and z-transforms
The (one-sided) Laplace transform (see e.g. [62]) is conveniently introduced for functions
y(t) defined only for t ≥ 0. It is a function of a complex variable s and given by:
Y (s) = L{y(t)} =
∫ ∞
t=0
y(t)e −st dt. (2.14)
Formally, it is defined for values of s for which the integral in (2.14) converges, which
constitutes the region of convergence. This usually is a half-plane. An important class
of functions used in signal processing and filtering involves sines, cosines, exponentials,
polynomials and from the non-standard functions the impulse functions, all of which
have rational Laplace transforms [62].
The continuous-time impulse function is the Dirac delta-function δ(t) is given by:
{
δ(t) = 0, ∀t ≠ 0
∫ ∞
−∞ δ(t)dt = 1 (2.15)
yielding an infinitesimally narrow, infinitely tall pulse which integrates to unity. On the
other hand the discrete-time impulse can be associated (via zero-order hold (ZOH)) with
a continuous time block function, called the Kronecker delta function:
{ δ[n] = 0, ∀n ≠ 0
(2.16)
δ[n] = 1, for n = 0.
Writing s = σ + iω, it is clear that the Fourier transform is obtained for σ = 0:
Y (iω) = Y (ω). (2.17)
(Laplace)
(Fourier)
Note that with abuse of notation we denote the Laplace transform of y(t) as Y (s), the
Laplace transform for s = iω as Y (iω), and the Fourier transform of y(t) as Y (ω).
As is well-known, for the Laplace transform Y (s) of a function y(t) the initial value
theorem:
y(0) = lim sY (s), (2.18)
|s|→∞

2.8. LINEAR SYSTEMS 19
and the final value theorem:
lim y(t) = lim sY (s), (2.19)
t→∞ s→0
hold.
In discrete time, a similar transform which is called the z-transform is used. Like
s for the Laplace transform, z is a complex variable. The z-transform of an (infinite)
sequence y[k], k ∈ Z is denoted by Z{y k } = Y (z) and is given by:
Y (z) = Z{y k } = ∑ k
y k z −k . (2.20)
Observe the similarity with the definition of the Laplace transform in (2.14). The z-
transform is the Laplace transform of an impulse train of a continuous function, with
sampling period T ∈ R + . Given this instantanious sampled function:
∞∑
n=−∞
y(nT )δ(t − nT ), (2.21)
where δ(k) is the Dirac delta as in (2.7). One can take the Laplace transform of this
sampled function and substitute z = e sT to obtain the z-transform of y(t). When taking
z = e iΩ in (2.20) the Fourier series as in (2.9) is obtained.
2.8 Linear systems
Let S be a linear system with input u and output y. Then the system S has the following
defining properties:
Homogeneity If the input u to a linear system S is scaled with a factor α, the output
of that system is scaled with a factor α as well: S(αu) = αS(u).
Superposition If two inputs are added and passed through a linear system, then the
output equals the sum of the outputs as if the two input signals were passed through
the system individually: S(u 1 + u 2 ) = S(u 1 ) + S(u 2 ).
Linear systems can be discrete-time or continuous-time, but in this section the focus
will be on continuous-time systems. Linear systems can also be time-invariant: for any
input u producing a corresponding output y = S(u), it holds that y τ = S(u τ ) for any
time shift τ, where u τ , y τ denote the signals u, y time shifted by τ. If a system is
both linear and time-invariant, one calls it a linear time-invariant system or LTI system
for short. The input/output relation of LTI systems can be represented by differential
equations. As an example of such an n th order differential equation takes the following
form:
y (n) (t) + a 1 y (n−1) (t) + . . . + a n−1 y (1) (t) + a n y(t)
= b 0 u (n) (t) + b 1 u (n−1) (t) + . . . + b n−1 u (1) (t) + b n u(t), (2.22)

20 CHAPTER 2. CARDIAC SIGNAL PROCESSING
where y (k) (t) denotes the k th derivative of y(t) with respect to t.
A system is causal if any input u with u(t) = 0, ∀t < 0 produces an output y for
which it holds that y(t) = 0, ∀t < 0.
LTI systems have an associated impulse response, step response and transfer function,
each characterizing the system. For a detailed discussion on these topics the reader is
referred to for example [62]. The impulse response function h(t) is defined as the output
of a system, corresponding to the Dirac delta δ(t) (2.15) as an input for continuous-time
systems, and the Kronecker delta δ[n] (2.16) for discrete-time systems.
The transfer function the Laplace transform of the impulse response. It is well known
that in the case of zero initial conditions the following relation holds for the Laplace
transform of the input U(s), the Laplace transform of the output Y (s) and the transfer
function H(s) of the LTI system:
Y (s) = H(s)U(s). (2.23)
The Laplace transform of δ(t) is: L {δ(t)} = D(s) = 1 so that for the Laplace transformed
output it holds that:
Y (s) = H(s)D(s) = H(s). (2.24)
If a causal LTI system is of finite order, it will possess a proper rational transfer function
[62] H(s):
H(s) = Y (s)
U(s) = b 0s n + b 1 s n−1 + . . . + b n
s n + a 1 s n−1 . (2.25)
+ . . . + a n
The roots of the numerator polynomial in (2.25) are called the zeros of the system and
the roots of the denominator polynomial the poles. The order n of the system is defined
as the McMillan degree of H(s). For single-input single-output (SISO) systems, the order
of the system equals the degree of the denominator of the transfer function H(s), after
canceling all common factors. In this work linear systems will be assumed to be finite
order, time-invariant and causal, unless stated otherwise.
Given a system with impulse response function h(t) and input u(t), the output y(t)
becomes the convolution of the input u(t) and the impulse response h(t):
y(t) = (h ∗ u)(t) =
∫ ∞
−∞
h(τ)u(t − τ)dτ. (2.26)
The impulse response function is the derivative of the step response function, and therefore
this latter one can be calculated from the transfer function as:
L −1 { 1
s H(s) }
. (2.27)
An LTI system is said to be asymptotically stable if the output corresponding to an
impulse input damps out to zero. Formally this comes down to the requirement that the
poles of its transfer function are all in the open complex left halfplane.
LTI systems can be represented in state-space:
ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t).
(2.28)

2.8. LINEAR SYSTEMS 21
The vector x(t) is the state vector. The n × n matrix A is called the system matrix, the
dynamical matrix or the state matrix, the n × 1 vector B is called the input vector, the
1 × n vector C the output vector and the scalar D is called the direct feedthrough term,
alias direct current or DC-term. A system may be denoted by this quadruple of matrices
S = (A, B, C, D).
We usually consider systems as input/output systems, hence focusing on its external
interactions. The system is identified with its transfer function. Its state-space representations
is merely used as a convenient “tool”, and having meaningful states in this
representation is not a requirement. To arrive at computationally convenient canonical
forms and parameterizations, similarity transforms are used. Given any non-singular
matrix T , a similarity transformation from a basis x(t) to a basis ˆx(t) = T x(t) may be
performed yielding an equivalent system:
where
˙ˆx(t) = Âx(t) + ˆBu(t)
y(t) = Ĉx(t) + Du(t), (2.29)
 = T AT −1 (2.30)
ˆB = T B (2.31)
Ĉ = CT −1 (2.32)
A canonical form is a “standard” form. We deal with a canonical form if for all
systems that are mutually equivalent to some given system, we transform it to the same
member of that equivalence class. As a result, in a canonical form different choices of the
parameters produce systems that are never equivalent. Using similarity transformations,
systems can be brought into a canonical form and from one canonical form to another.
For state-space systems the order n has the interpretation that it is the state-space
dimension that is minimally required to represent the system.
From the state-space representation S = (A, B, C, D) the transfer function can be
determined:
H(s) = D + C(sI − A) −1 B. (2.33)
The poles of the system correspond to the eigenvalues of the system matrix A. As
one would expect from the above it is also possible to compute the impulse response
associated with the system:
h(t) = Ce At B + Dδ(t), t ≥ 0. (2.34)
Insted of a SISO system a multi-input multi-output system (MIMO) is considered
with p inputs and q outputs. The vectors B and C and the scalar D change accordingly.
B then becomes an n × p input matrix, C a q × n output matrix and D a q × p direct
feedthrough. All other formulas previously discussed in this section still apply.

22 CHAPTER 2. CARDIAC SIGNAL PROCESSING
For discrete-time systems similar relations and properties as for continuous-time systems
hold. The state-space description in (2.28) takes the form:
x[k + 1] = Ax[k] + Bu[k]
y[k] = Cx[k] + Du[k].
(2.35)
The transfer function can be obtained from the state-space representation similarly as in
(2.33) as
H(z) = D + C(zI − A) −1 B, (2.36)
and (2.34) changes to:
h[k] =
{ D k = 0
CA k−1 B k > 0.
(2.37)
For a more in-depth discussion on linear systems and their properties the reader is
referred to the relevant literature, for example [62, 115].
2.9 Filtering signals
It is well-known that a sinusoid u s (t) = sin(ωt) used as input for a stable linear system
with transfer function H(s) yields a steady state output y s (t) as:
y s (t) = A sin(ωt + φ), (2.38)
where A = |H(iω)| is that absolute value and φ = ∠H(iω) is the angle of the complex
number H(iω) in the complex plane. The function H(iω) is called the frequency response
function of the system and is the Fourier transform of the impulse response h(t). Equation
(2.38) is called the property of sinusoidal fidelity. This property implies that if a sinusoid
is used as an input to such a system, then a sinusoid of the same frequency will constitute
the output. However the output sinusoid exhibits a phase shift φ relative to the input
sinusoid and an amplitude gain of |H(iω)|. If the phase response of the LTI system is
linearwith the frequency, it is said that the system has linear phase. In a linear phase
system all frequencies have equal delay times, so that there is no phase distortion.
As discussed in Section 2.6, each signal can be decomposed in terms of sines and
cosines. Consequently, if a Bode plot of the frequency response of the system is considered,
one can examine how each frequency in an input u(t) signal is affected by the
system. How each frequency is affected by the system depends on the placement of the
poles and zeros in the complex s-plane: frequencies near zeros will be suppressed and
frequencies near poles will be amplified. However, if poles and zeros are sufficiently close
there will be interaction between the poles and zeros. By appropriate placement of the
poles different type of filters can be created such as low-pass, high-pass, band-pass and
notch filters. These can have properties such as fast roll-off, no pass/stop-band ripple,
etc. Classically, these filters are designed to have certain properties in the Fourier
domain.

2.9. FILTERING SIGNALS 23
It is useful to consider filtering in the time-domain. From (2.26) it is clear that if
a signal u(t) is used as an input, it is convoluted with the impulse response h(t) of the
system, which is the inverse Laplace transform of the transfer function.
A similar property holds in discrete time. For a discrete-time signal that is a sampled
version of a continuous-time signal, as discussed on page 15, meaning can be given to
the sampling frequency f in Hertz. Since sensor data is mostly continuous-time data, for
most applications this f is important to relate to the frequencies in the source signal. For
applications where the data is inherently discrete-time one can choose f = 1. Given an
(
input u s [k] = sin
H(z) is:
ωk
f
)
, the corresponding output y s [k] of a system with transfer function
[ ] ωk
y s [k] = A sin
f + φ , (2.39)
where where A = |H ( e ) iω/f | and φ = ∠H ( e ) iω/f . The function H ( e ) iω/f is the
discrete-time frequency response. Frequencies in input signals are thus similarly affected
as in the continuous-time case.
As in (2.20) let u(z) = u 0 + u 1 z −1 + u 2 z −2 + . . . be the z-transform of a discretetime
signal u = {u 0 , u 1 , u 2 , . . .}. For discrete-time filters with a proper rational transfer
function of the following form:
H(z) = b 0z n + b 1 z n−1 + . . . + b n
z n = b 0 + b 1 z −1 + . . . + b n z −n , (2.40)
the z-transformed output corresponding to a transformed input u(z) can be computed
as:
H(z)u(z) = b 0 u 0 + (b 0 u 1 + b 1 u 0 )z −1 + (b 0 u 2 + b 1 u 1 + b 2 u 0 )z −2
+ (b 0 u 3 + b 1 u 2 + b 2 u 1 + b 3 u 0 )z −3 . . . (2.41)
The impulse response u 0 = 1, u k = 0 for k > 0 can be easily read off from this expression
yielding {b 0 , b 1 , . . . , b n , 0, 0, . . .} and will have a finite length, i.e., h[k] = 0, ∀k > n. This
system is said to be of finite impulse response (FIR). Since in linear systems the input
is convoluted with the impulse response, such systems exhibit a convenient one-to-one
correspondence of the transformation from input to output in the time and z-domain.
The output at each time instance is the weighted mean of the input around that instance,
where the weighing factors are defined by the numerator polynomial of the transfer
function, hence the name moving average (MA) filters. The filter banks associated with
the discrete wavelet transform in Section 3.2 take such a form (2.40).
Historically filters are designed based on frequency domain properties. However in
the case of in-band noise, such as muscle artifact noise in ECGs, the signal is not effectively
separated from the noise in the frequency representation and consequently the
noise cannot be removed using this approach. However the noise may have a different
morphology from that of the signal and filtering techniques that take morphology into
account, such as wavelets, may be beneficial.

Chapter 3
Wavelet transformations
Nowadays, a quarter of a century after Morlet and Grossman first coined the word
“wavelet” and formalized the corresponding transform [44], wavelets are recognized as a
fundamental and powerful tool in signal analysis, and widely applied in practice, including
in the field of biomedical engineering. The remarkable rise and success of wavelets,
starting from the 70s, may largely be attributed to its added value to Fourier analysis.
The wavelet transform offers both time and frequency localization, whereas the Fourier
transform only offers frequency information, making the wavelet transform a powerful
tool for signal analysis [79, 107]. The windowed Fourier transform (2.13) offers the
ability to obtain both time and frequency information too, but this method does not
have the flexibility of the wavelet transform since the basis is restricted to sines and
cosines (harmonics) and it does not possess the “zoom-in” property of wavelets as discussed
on page page 26. In the late 1980s Yves Meyer published work on orthogonal
wavelets (e.g.[85]). This inspired Stéphane Mallat to construct a multi-resolution theory
on wavelets [79], whereas Ingrid Daubechies constructed a well-known set of orthonormal
wavelets [30]. Despite its recent invention, wavelet theory is already well established and
employed in many standards such as the jpeg standard for image compression.
Many biomedical applications have been developed around wavelets. For example
the detection of characteristic points in ECGs [105, 61, 78, 83, 102, 2], filtering [98, 112],
separation of fetal cardiac activity [87] and many others. In [101] it has been shown
that wavelets can be finetuned for a given application. Due to the work in [107, 92]
on orthogonal filter banks a parameterization is available for scalar wavelets that can be
used as a framework for wavelet design. However no optimization criterion was discussed.
In Chapter 5 an optimization criterion is introduced for the design of optimal wavelets.
In this chapter we will briefly introduce some basic concepts of wavelets, and then
focus in some depth on the relation between wavelets and filter banks. This chapter will
close with a discussion on multiwavelets and their relation with lossless systems.
25

26 CHAPTER 3. WAVELET TRANSFORMATIONS
3.1 Continuous wavelets
The wavelet transformation if aimed at the decomposition of a signal, localized in both
time and frequency simultaneously by inducing a change of basis for the signal in question.
In order to accomplish this a so called “wavelet” function is used as a basis, and this
wavelet function has the property that it is localized in both time and frequency. More
formally, the wavelet transformation of a real signal x(t) using a real wavelet ψ(t) is
defined as the cross-covariance at lag τ of that signal with the normalized dilated wavelet
1 √ σ
ψ( t σ
) (see for example [44, 80]):
W (τ, σ) =
∫ ∞
−∞
x(t) 1 √ σ
ψ
( t − τ
σ
)
dt, τ ∈ R, σ ∈ R + . (3.1)
The parameter σ is a scale parameter that determines the (pseudo)frequency localization
of the wavelet transform and the parameter τ determines the time localization of the
wavelet transform. For a specific scale σ and a specific time τ the L 2 -inner product
1
between the signal x(t) and the normalized, time-shifted and scaled wavelet √σ ψ ( )
t−τ
σ
is thus computed. The form in (3.1) is known as the continuous wavelet transform (CWT)
or as the integral wavelet transform. The continuous wavelet transform is an invertible
transform as shown below:
x(t) = 1 ∫ ∞ ∫ ∞
W (τ, σ) 1 ( ) t − τ
√ ψ dτ dσ
C
σ σ σ 2 , t ∈ R, C ∈ R+ , (3.2)
0
−∞
where C is a constant that depends on the wavelet ψ(t) at hand. Not any arbitrary function
can be used as a wavelet basis and therefore the function ψ(t) must obey a so-called
admissability condition [44]. The admissibility condition comes down to requiring that
ψ(t) has zero average and that its Fourier transform Ψ(ω) is continuously differentiable
[80]. For a more detailed discussion on this subject the reader is referred to for example
[80, 26].
We denote dilated and time-shifted wavelet functions as
ψ τ,σ (t) = 1 √ σ
ψ
Their corresponding Fourier transform is:
( t − τ
σ
)
. (3.3)
Ψ τ,σ (ω) = e −iτω Ψ(σω). (3.4)
From (3.4) it can be seen that unlike in (2.13) where the Heisenberg uncertainty rectangle
has constant dimensions in the time-scale plane. In this case when moving to the coarser
scales (i.e. lower frequencies) the window tightens in the scale dimension and widens in
the time dimension. Conversely the window widens in the scale dimension and tightens
in the time dimension when moving to finer scales (higher frequencies). The dimensions
of the uncertainty rectangle remain constant when moving it along the time dimension.
This behavior of the uncertainty rectangle facilitates the “zoom-in” property of wavelets.

3.2. WAVELETS FROM FILTER BANKS 27
0.8
0.6
0.4
Gaussian wavelet
Mexican Hat wavelet
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−5 0 5
Figure 3.1: The Gaussian and the Mexican Hat wavelet
In cardiac signal processing the Gaussian wavelet ψ(t) = −2( 2 π ) 1 4 te −t2 is very popular
(see for example [5, 102]), partly due to the fact that the wave-like shape as shown in
Figure 3.1 resembles the QRS complex, partly due to some of its other nice properties
such as the fact that it is a derivative of a smoothing function [80]. The Gaussian wavelet
is constructed by taking the normalized derivative of the Gaussian probability density
function (pdf) with mean zero and variance 0.5. Higher order derivatives of the Gaussian
pdf can be taken to obtain Gaussian wavelets of various orders such as the seconde
2
derivative of the Gaussian pdf: the Mexican Hat wavelet ψ(t) =
π 0.25√ 3 (t2 − 1)e −0.5t2
which is a very popular wavelet for ECG processing [2, 20] too. When the question is
posed what wavelet to use, the answer depends heavily on the application and signals at
hand. Considering the large variety of functions that obey the admissability conditions
there is a good chance that one can do better than by choosing a popular and well-known
wavelet.
3.2 Wavelets from filter banks
The wavelet transform can also be performed on discrete time signals, thereby still using
a continuous wavelet function ψ(t). One way to do this is with discrete filters from
filter banks as in the discrete wavelet transform (DWT). In this thesis the framework of
DWT wavelets from filter banks as described in for example [107] is used as a setting for
wavelet design. This framework has a number of advantages and restrictions that make
it a convenient setting for wavelet design. Filter banks may have, when imposed, five
important properties of interest to us, which are further addressed below:
• Perfect reconstruction
• Orthogonality of the filter bank and the underlying wavelet based multi-resolution
structure

28 CHAPTER 3. WAVELET TRANSFORMATIONS
Analysis bank
Synthesis bank
x(z)
H 0 (z)
H 0 (z)x(z)
2
a(z)
2
½(H 0 (z)x(z)+
H 0 (-z)x(-z))
F 0 (z)
H 1 (z)
H 1 (z)z(z)
2
b(z)
2
½(H 1 (z)x(z)+
H 1 (-z)x(-z))
F 1 (z)
+
z -N x(z)
Figure 3.2: Wavelet analysis and synthesis
• Flatness of the filters and vanishing moments in the wavelets
• Smoothness of the wavelets
• Linear phase
In discrete-time the input sequence is processed by two filters in parallel that split
the input sequence in terms of frequency. These filters relate to the wavelet function
in such a way that the downsampled output corresponds to the filtering of the signal
with the wavelet function at a particular scale. The discrete-time input signal {x n } is
essentially fed through a low-pass filter H 0 (z) and a high-pass filter H 1 (z) in parallel
[38]. The filters H 0 (z) and H 1 (z) form a filter bank. The pair of filters in the filter bank
are designed in such a way that after downsampling by a factor 2 no information is lost
and there is no redundancy; they are critically sampled. In formulas, downsampling with
a factor two will be written as ↓2. The output of the low-pass channel is a sequence of
approximation coefficients (alias scaling coefficients) and the output from the high-pass
channel a sequence of detail coefficients (alias wavelet coefficients). The original input
can be reconstructed by upsampling and filtering each channel with the corresponding
synthesis filter F 0 (z) or F 1 (z) and adding the two channels as illustrated in Figure 3.2.
The downsampling can be implemented with frequency modulation [107]:
a(z) =↓2H 0 (z)X(z) = 1 2
(
H 0 (z 1 2 )X(z
1
2 ) + H0 (−z 1 2 )X(−z
1
2 )
)
. (3.5)
The aliasing term (with −z) cancels the odd part. This works similarly for the detail
coefficients b(z). To upsample a(z) again take a(z 2 ), which comes down to inserting a 0
after each element in the sequence.
Example 3.2.1. As an example consider the Haar wavelet with low-pass filter H 0 (z) =
1
2 + 1 2 z and high-pass filter H 0(z) = − 1 2 + 1 2z. For input x = {4, 6, 8, 7} the low-pass
output will be a = {5, 7.5} and the high-pass output b = {−1, 0.5}.

3.2. WAVELETS FROM FILTER BANKS 29
3.2.1 Perfect reconstruction
From this point on H 0 (z) and H 1 (z) are assumed to be FIR (finite impulse response)
filters:
H 0 (z) = c 0 + c 1 z −1 + . . . + c N z −N , (3.6)
H 1 (z) = d 0 + d 1 z −1 + . . . + d N z −N . (3.7)
This will result in wavelets that are compactly supported [107] and makes the conditions
in this paragraph easy to satisfy.
To have perfect reconstruction it means that if a signal is decomposed with an analysis
filter bank and then reconstructed with a corresponding synthesis filter bank, the
output signal is equal to the input signal expect for a possible delay. To ensure perfect
reconstruction there may be no aliasing and no distortion.
The aliasing effect [38] is present in both channels and must be canceled by the
synthesis filters F 0 and F 1 :
F 0 (z)H 0 (−z) + F 1 (z)H 1 (−z) = 0. (3.8)
To ensure that there is no distortion in the output of Figure 3.2, the following condition
must hold:
F 0 (z)H 0 (z) + F 1 (z)H 1 (z) = 2z −Q , (3.9)
where Q is the overall delay of the filter bank [107].
The conditions (3.8) and (3.9) can be written in terms of a single modulation matrix:
( ) ( )
F0 (z) F 1 (z) H0 (z) H 0 (−z)
=
F 0 (−z) F 1 (−z) H 1 (z) H 1 (−z)
3.2.2 Orthogonal filter banks
( )
2z
−Q
0
0 2z −Q . (3.10)
For orthogonal filter banks it holds that the impulse responses of the synthesis filters are
the time reverses of the impulse responses of the analysis filters [106] as expressed by
(3.13). To see this, we start by observing that the two filters in the filter bank H 0 (z)
and H 1 (z) form a power complementary set [117, page 183]. Since on the unit circle
the complex conjugate of z is z −1 the orthogonality condition from [117], taking the
differences in notation into account, comes down to:
H 0 (z −1 )H 0 (z) + H 1 (z −1 )H 1 (z) = c, (3.11)
where c = 2 for consistency with (3.9), that is, to ensure conservation of energy.
In this orthogonal framework the condition (3.8) can be enforced by constructing the
synthesis filters from the analysis filters by alternating signs [31]:
F 0 (z) = H 1 (−z) and F 1 (z) = −H 0 (−z). (3.12)

30 CHAPTER 3. WAVELET TRANSFORMATIONS
To satisfy the condition in (3.9) the power complementary property can be used [117]:
F 0 (z) = z −N H 0 (z −1 ) and F 1 (z) = z −N H 1 (z −1 ), (3.13)
with the result that for orthogonal filters it holds that in (3.9) Q = N.
To satisfy both (3.12) and (3.13) for an N th order filter one can construct the highpass
filter from the low-pass filter by the alternating flip construction [31]. From the left
side of (3.12) and (3.13) it is obtained that:
After substituting z → −z it is obtained that:
H 1 (−z) = z −N H 0 (z −1 ). (3.14)
H 1 (z) = (−z) −N H 0 (−z −1 ). (3.15)
And from the right side of (3.12) and (3.13) it is obtained that:
After substituting z → z −1 it is obtained that:
H 1 (z −1 ) = z N F 1 (z) = −z N H 0 (−z). (3.16)
H 1 (z) = −z −N H 0 (−z −1 ). (3.17)
From (3.15) and (3.17) it follows that N must be odd yielding the alternating flip construction:
H 1 (z) = (−z) −N H 0 (−z −1 ), for N = 2n − 1. (3.18)
In terms of (3.6) and (3.7) with N = 2n − 1 it follows that:
d k = (−1) k c N−k (k = 0, 1, . . . , N). (3.19)
Filters that are constructed in this manner are called quadrature mirror filters [38, 86, 30].
For orthogonality the following constraints hold for the scaling filter coefficients c k
and the wavelet filter coefficients d k of H 0 (z) and H 1 (z) respectively:
Normalization ∑ N
k=0 c2 k = 1 and ∑ N
k=0 d2 k = 1
Double-shift orthogonality ∑ N
k=0 c kc k−2l = 0 and ∑ N
k=0 d kd k−2l = 0 ∀l ∈ Z\ {0}
Double-shift orthogonality between the filters ∑ N
k=0 c kd k−2l = 0 ∀l ∈ Z
In the last two conditions negatively indexed coefficients are all zero by convention (c 0 ,
d k for k < 0 or k > N).

3.2. WAVELETS FROM FILTER BANKS 31
Level j+1
a (j-1)
Level j
H 0 2
H 1 2
a (j)
b (j)
H 0 2
H 1 2
a (j+1)
b (j+1)
Figure 3.3: Wavelet analysis tree structure
3.2.3 Multi-resolution analysis
The frequency localization for the discrete wavelet transform comes from using dilated
basis functions, known as the scaling and wavelet function, as will be discussed in Section
3.2.4. However this multi-resolution analysis can be implemented by a cascade of filter
banks, consisting of an orthogonal pair of low- and high-pass filters. The output of the
low-pass filter is used as the input of the next level as in Mallat’s algorithm [79]. This
procedure is illustrated in Figure 3.3 and gives rise to multi-resolution analysis (MRA).
Note that we now have approximation and detail coefficients at various levels j that
correspond to dyadic scales 2 j .
The basic principle is as follows: One starts with a signal x and feeds it through an
analysis filter bank. The output of the low-pass filter is then a (1) =↓2H 0 x and that of
the high-pass filter is b (1) =↓2H 1 x. Next a (1) is used as the input of the filter bank to
yield a (2) =↓2H 0 a (1) and b (2) =↓2H 1 a (1) . Then again repeating this procedure a number
of times, the output of the low-pass filter is used so that one eventually ends up with
{b (1) , b (2) , . . . , b (L) , a (L) }. The signal x can be reconstructed by a cascade of synthesis
filter banks. The length of the signal x and the size of the filter N determines how many
levels this cascade can have. If the maximum number of levels L has been reached, the
signal is said to be fully decomposed.
3.2.4 Wavelet and scaling functions
In this section it will be discussed how filter banks can be interpreted using the wavelet
paradigm. For an excellent exposition on this topic the reader is referred to [107].
Consider the following multi-resolution structure for the L 2 -space, consisting of a
nested sequence of linear subspaces V l , called approximation spaces:
. . . ⊂ V 1 ⊂ V 0 ⊂ V −1 ⊂ . . . , (3.20)
with ⋂ l∈Z V l = {0} and ⋃ l∈Z V l = L 2 (R), where the bar indicates closure. Assume
that there exists a function ϕ(t), called the scaling function, that spans the subspace V 0 .

32 CHAPTER 3. WAVELET TRANSFORMATIONS
This scaling function generates a shift invariant orthonormal basis {ϕ(t − k)|k ∈ Z} of
V 0 . From V 0 the orthonormal bases are induced for all spaces V l [80]:
ϕ (l)
k (t) = √ 1 ( t − 2 l )
ϕ k (
2
l 2 l = 2 −l
2 ϕ 2 −l t − k )
V l = span{ϕ (l)
k
|k ∈ Z}. (3.21)
Furthermore, consider the subspaces W l called detail spaces, that are the orthogonal
complements of the approximation spaces V l :
V l ⊕ W l = V l−1 . (3.22)
Assume that the approximation space W 0 is spanned by a the wavelet function ψ(t), and
its dilated versions are shift-invariant orthonormal bases for the spaces W l :
ψ (l)
k (t) = √ 1 ( t − 2 l )
ψ k (
2
l 2 l = 2 −l
2 ψ 2 −l t − k )
W l = span{ψ (l)
k
|k ∈ Z}. (3.23)
From (3.20) and (3.21), it follows that there exists a sequence of coefficients {c k } such
that the scaling function can be written as a linear combination of dilated versions of
itself in the dilation equation:
ϕ(t) = √ 2
N∑
c k ϕ(2t − k). (3.24)
k=0
Likewise from (3.20), (3.22) and (3.23), it follows that there exists a sequence of
coefficients {d k } such that the wavelet function can be written as a linear combination
of dilated versions the scaling function in the wavelet equation:
ψ(t) = √ 2
N∑
d k ϕ(2t − k). (3.25)
k=0
The coefficients c k and d k in (3.24) and (3.25) are the same coefficients as in Section
3.2.2, with all properties involved, linking the filter coefficients to the wavelet and
scaling function. This allows us to reverse the approach and given coefficients c k and
d k try to find a scaling function ϕ(t) and a wavelet function ψ(t). An iteration scheme
which allows for the exact computation of ϕ(t) at all the dyadic points up to an arbitrary
resolution can be found, for instance, in [107, section 6.1]. We have adopted this
method in this work. It is important to note that it may well happen that the resulting
sequence exhibits a discontinuous and fractal structure and may not converge to an actual
function, i.e., ϕ(t) and ψ(t) may not exist for a given set of filters. The continuity
of the wavelet and scaling functions is further investigated in terms of the “joint spectral
radius” in [110, Section 2.2].

3.2. WAVELETS FROM FILTER BANKS 33
Now let f(t) be a function, corresponding to a sequence {x k } of wavelet coefficients,
that can be expressed in terms of the wavelet basis for V 0 as:
f(t) = ∑ k
x k ϕ(t − k), (3.26)
then the coefficients resulting from the DWT give an expansion for f(t) in terms of the
orthonormal functions ϕ (l)
k
(t) and ψ(l)
k
(t):
f(t) = ∑ k
a (L)
k
ϕ(L) k
L (t) + ∑ ∑
where L is no more than the maximum level as in Section 3.2.3.
wavelet structure it then also holds [91] that:
a (l)
k
=
b (l)
k
=
∫ ∞
−∞
∫ ∞
−∞
l=1
k
b (l)
k
ψ(l) k
(t), (3.27)
For an orthogonal
ϕ (l)
k
(t)f(t)dt, (3.28)
ψ (l)
k
(t)f(t)dt, (3.29)
which connects the continuous wavelet transform in (3.1) and discrete wavelet transform
(3.29). Note that with the discrete wavelet transform these expansion coefficients {a (l)
k }
and {b (l)
k
} can be calculated without the explicit use of the scaling and wavelet function,
but only by making use of a filter bank, using the approach in Section 3.2.3.
3.2.5 Vanishing moments
For some applications it is beneficial that the wavelets have vanishing moments. If a
wavelet has p vanishing moments then the wavelet function ψ(t) is orthogonal to polynomials
of up to degree p − 1 [80]. This leads to the following equation, which is familiar
for moments from physics and statistics:
∫ ∞
−∞
t k ψ(t)dt = 0, for 0 ≤ k < p. (3.30)
Due to the orthogonality between the wavelet and scaling function it follows that if a
wavelet has p vanishing moments then the space V 0 = span{ϕ k |k ∈ Z} spanned by
the scaling function contains all the polynomials of degree less then p [107, Section
7.1]. As familiar from Taylor series, smooth functions can locally be approximated by
polynomials. If a wavelet has p vanishing moments, then the detail signal will contain no
energy of polynomials up to degree p − 1. If a signal f(t) can locally in neighborhood v
be represented as a polynomial part f v,p (t) with degree less then p and some error term
ε v (t) then the wavelet transform of f will be [80, Section 6.1]:
∫ ∞
W fv (τ, σ) = f v (t) 1 ( ) ∫ t − τ
∞
√ ψ dt = ɛ v (t) 1 ( ) t − τ
√ ψ dt = W ɛv (τ, σ).
−∞ σ σ
−∞ σ σ
(3.31)

34 CHAPTER 3. WAVELET TRANSFORMATIONS
When attempting to locally approximate the signal f with polynomials of degree less
then p, ɛ represents the approximation error. The approximation error will have a nonzero
contribution to the detail coefficients. This property can be used to measure the
Lipschitz regularity (also known as the Hölder exponent) of a signal [80, Section 6.1].
The approximation error decays with the level j (scale 2 j ) and the number of vanishing
moments as O(2 jp ) [107, Section 7.1].
For discrete time signals and wavelets, vanishing moments have to be imposed on the
impulse response of H 1 (z), that is, moments up to order p − 1 have to be imposed on the
wavelet filter. To have one vanishing moment amounts to the condition d 0 +d 1 +. . .+d N =
0 which is equivalent to the commonly imposed condition that the integral of the mother
wavelet ψ(t) is equal to zero: ∫ ∞
ψ(t)dt = 0. In Section 3.2.4 it is discussed how the
−∞
wavelet function is obtained from the filter coefficients. For a wavelet filter to have p
vanishing moments the constraints on the high-pass coefficients are:
N∑
l k d l = 0, for 0 ≤ k < p. (3.32)
l=0
The conditions on the wavelet filter (3.32) to have p vanishing moments can be translated
to the scaling and wavelet function using the dilation equation (3.24) and the wavelet
equation (3.25). From this it follows that the conditions on the filter are a linear combination
of the conditions on the wavelet function (3.30) to have p vanishing moments,
vice versa.
For a wavelet filter to have vanishing moments, it is required that H 0 (z) and H 1 (z)
have zeros at z = −1 and z = 1 respectively [80, Theorem 7.4][107, Theorem 7.1]. For
filters this corresponds to requiring zeros at respectively π radians (Nyquist frequency)
for the scaling filter and 0 radians for the wavelet filter. The zeros for the scaling filter at π
radians make it a low-pass filter with a corresponding degree of flatness and the wavelet
filter a high-pass filter with a certain degree of flatness. The Daubechies orthogonal
wavelet family is constructed by using all freedom to impose vanishing moments.
3.2.6 Linear phase
As discussed on page page 22, linear phase helps to avoid phase distortion. There only
exists a single orthogonal wavelet with linear phase: the Haar wavelet (see e.g. [120]).
In order to obtain both orthogonality and linear phase simultaneously, the theory of
multiwavelets (see Section 3.4) can be employed.
3.2.7 Polyphase filtering
In the traditional discrete wavelet transform approach,the high- and low-pass filters operate
at full rate, after which the outputs are downsampled. In other words: half of the
processed information is discarded. The use of polyphase filters avoids this, by first splitting
the input into two or more (M) phases, then applying a polyphase filter to each phase
separately and finally combining the polyphase outputs to generate the intended output.

3.2. WAVELETS FROM FILTER BANKS 35
X(z)
2
X e
(z)
H 0,e
(z)
z -1
2
z -1 X o
(z)
H 0,o
(z)
+ (H 0 (z)X(z)) e
Figure 3.4: Polyphase low-pass filter
The polyphase filters operate at reduced rate 1 M
. Suppose an input signal X(z) and an
analysis low-pass filter H 0 (z). The goal is to obtain ↓2H 0 X; the downsampled result of
filtering X(z) with H 0 . To implement this with polyphase filters, with e.g., M = 2, take
H 0,e (z) = c 0 +c 2 z −1 +. . .+c 2n−2 z −(n−1) and H 0,o (z) = c 1 +c 3 z −1 +. . .+c 2n−1 z −(n−1) as
respectively the even and odd phase of the filter H 0 (z), and X e (z) and X o (z) as respectively
the even and odd phase of the input signal X(z). (Since M = 2 it is convenient to
call one of the phases “even” and the other one “odd”.) Then the desired output ↓2H 0 x
can be calculated [107, Section 4.2] with polyphase filters as:
Z {↓2H 0 x} = (H 0 (z)X(z)) e
= X e (z)H 0,e (z) + z −1 H 0,o (z)X o (z)
= [ H 0,e (z) H 0,o (z) ] [ ]
X e (z)
z −1 ,
X o (z)
(3.33)
This process is illustrated in Figure 3.4. The basic idea behind this approach to find
the even part, magister dixit, “So it is really odd plus odd, and even plus even” [107, p.
115].
Now consider a filter bank with low-pass filter H 0 and high-pass filter H 1 . These
filters can be split into two phases and a polyphase matrix H p (z) can be constructed as:
[ ]
H0,e (z) H
H p (z) =
0,o (z)
, (3.34)
H 1,e (z) H 1,o (z)
Then this polyphase matrix H p (z) can be used to implement the low- and high-pass filter
in parallel by acting on the same polyphase input:
[ ] [ ]
V0 (z) (H0 (z)X(z))
=
e
=
V 1 (z) (H 1 (z)X(z)) e
[ ] [ ]
H0,e (z) H 0,o (z) Xe (z)
H 1,e (z) H 1,o (z) z −1 . (3.35)
X o (z)
For reconstruction the synthesis filter can be implemented as a polyphase filter F p (z)

36 CHAPTER 3. WAVELET TRANSFORMATIONS
X(z)
2
X e (z)
V 0 (z)
H
z -1 p (z) F p (z)
X o (z)
V 1 (z)
2
F 0,o (z)V 0 (z)
+F 1,o V 1 (z)
F 0,e (z)V 0 (z)
+F 1,e V 1 (z)
2
2 +
Z -1
^
X(z)
Figure 3.5: Polyphase analysis and synthesis filters
too:
F p (z) =
[
F0,o (z)
]
F 1,o (z)
F 0,e (z) F 1,e (z)
ˆX(z) = [ z −1 1 ] F p (z 2 )
[
V0 (z)
V 1 (z)
(3.36)
]
. (3.37)
In the polyphase approach the analysis and synthesis filters are adjacent, in contrast to
the filterbank approach where they are separated by down- and upsampling. As a result
the condition for perfect reconstruction becomes elegant and simple for polyphase filters:
F p (z)H p (z) = z −q I. (3.38)
For practical purposes, the delay q should be as small as possible. The overall delay
induced by the filter system due to the fact that the filters are causal is Q = 2q + 1,
so that ˆX(z) = z −Q X(z). In the case of orthogonality as in Section 3.2.2 it holds
that Q = N = 2n − 1. The elegant condition in (3.38) facilitates the parameterization
of wavelet filters. Due to the fact that the polyphase filters operate at half rate this
approach is also beneficial from a computational viewpoint.
3.3 The stationary wavelet transform
The multi-resolution approach discussed in Section 3.2.3 is well suited for compression
purposes since it is critically sampled, i.e., all resulting wavelet coefficients are required
to ensure perfect reconstruction without redundancy. If a sequence x of length 2 m is
filtered with wavelet filters H 0 and H 1 , then x is mapped to ↓2H 0 x and ↓2H 1 x by means
of a projection. Both ↓2H 0 x and ↓2H 1 x will then have a length of approximately 2 m−1 ,
effectively reducing the resolution by a factor two. After rewriting the upper right-hand
side of (3.23) as
ψ (l)
−l
k
(t) = 2 2 ψ(2 −l (t − 2 l k))
= 2 −l
2 ψ(2 −l (t − τ)), τ = 2 l k, k ∈ Z, (3.39)
one can see that the timeshifts of the dilated wavelet functions 2 −l
2 ψ(2 −l t) are integer
multiples of 2 l with an initial offset of zero.

3.3. THE STATIONARY WAVELET TRANSFORM 37
When employing the wavelet transform for detection purposes, this approach has two
important drawbacks, in casu the lack of shift invariance and the loss of resolution at
coarse scales. These problems can be avoided by using an overcomplete wavelet transform.
The so-called stationary wavelet transform [91] achieves this by taking timeshifts equal
to all integer values instead of 2 l as in (3.39).
From (3.29) the connection between the continuous and discrete wavelet transform
becomes clear. If (3.29) is applied at an appropriate resolution at each scale, one obtains
a transform with the same resolution as the original signal x of length n = 2 m . The
regular DWT uses downsampling at each successive scale, which causes loss of resolution
at those scales. One way to avoid this is to perform the DWT 2 L times (L being the
number of scales in the MRA) and each time shifting the sequence x by one.
The results can then be combined to have full resolution at each scale; all the shifted
wavelet transforms combined give a shift invariant, redundant transform. A major drawback
of this approach is that it is very inefficient, since such a large number of shifts are
not required at fine scales.
Another approach is to split the wavelet decomposition at each scale. Let a (l−1) be
the input to scale l. Then one can perform a wavelet transform on the approximation
signal at the preceding level a (l−1) and on its time-shifted version z −1 a (l−1) . This results
in a tree that branches out exponentially fast compared to the the decomposition tree in
the regular discrete wavelet transform
a (l),0 =↓2H 0 a (l−1)
b (l),0 =↓2H 1 a (l−1)
a (l),1 =↓2H 0 z −1 a (l−1)
b (l),1 =↓2H 1 z −1 a (l−1) .
(3.40)
This process is also illustrated in Figure 3.6. Due to recursion we get
a (l),[0,S] =↓2H 0 a (l−1),S , (3.41)
where S is a bit string that encodes through what downsampling types (even or odd)
the signal has passed, with the most recent type up front. A 0 in S indicates that the
input passed through a downsampling ↓ 2, i.e., the regular detail and approximation
coefficients are involved, and an 1 indicates that it has passed through a delay followed
by downsampling ↓ 2z −1 , i.e., the detail and approximation coefficients of the shifted
signal are involved.
At scale k (scale k = 0 corresponds to the signal) there are thus 2 k sets of approximation
and detail coefficients, with each a associated bit string. The sets of detail
coefficients at each scale have to be interlaced such that a single set of coefficients is

38 CHAPTER 3. WAVELET TRANSFORMATIONS
a (l-1),s
a (l),s
H 0 2
a (l),[0,s]
H 0
a (l+1),[0,s]
H 1
b (l),s
2
z -1 2
z -1 2
a (l),[1,s]
H 1
H 0
b (l+1),[0,s]
a (l+1),[1,s]
2
z -1 2
H 1
b (l+1),[1,s]
a (l+1),[0,0,s]
a (l+1),[1,0,s]
a (l+1),[0,1,s]
a (l+1),[1,1,s]
Figure 3.6: Stationary wavelet transform
acquired at full resolution. For example at level l + 1 we have:
b even (l+1),[0,S] = b (l+1),[0,0,S] , (3.42)
b (l+1),[0,S]
odd
= b (l+1),[1,0,S] , (3.43)
b even (l+1),[1,S] = b (l+1),[0,1,S] , (3.44)
b (l+1),[1,S]
odd
= b (l+1),[1,1,S] , (3.45)
b (l+1),S
even = b (l+1),[0,S] , (3.46)
b (l+1),S
odd
= b (l+1),[1,S] . (3.47)
This process is displayed in Figure 3.7. For the maximum level L the sets of approximation
coefficients also have to be interlaced, likewise to obtain a wavelet decomposition
with full resolution at each dyadic scale.
The stationary wavelet transform can also be calculated in polyphase. The stationary
wavelet transform in polyphase representation is partially displayed in Figure 3.8, where
H p is a polyphase filter with two inputs (the phases) and two outputs.
Another approach to calculate the stationary wavelet transform is to modify the
filters. Such an approach, which gives the same results, is discussed in [91].

3.3. THE STATIONARY WAVELET TRANSFORM 39
b (l+1),[0,0,s]
b (l+1),[1,1,s] 2 z
2
b (l+1),[1,0,s]
2 z
b (l+1),[0,1,s]
2
+
+
b (l+1),[0,s]
b (l+1),[1,s]
2
2 z +
b (l+1),s
Figure 3.7: Interlacing for the stationary wavelet transform
a (l-1),s
2
a (l),[0,s]
z -1 2
H p
b (l),[0,s]
2
a (l),[1,s]
z -1 H p
b (l),[1,s]
2 z +
b (l),s
Figure 3.8: Stationary wavelet transform in polyphase representation

40 CHAPTER 3. WAVELET TRANSFORMATIONS
3.4 Multiwavelets
Regular scalar wavelets have a number of limitations. For example both linear phase and
orthogonality is only possible for the Haar wavelet as discussed in Section 3.2.6. Another
problem is that for a given choice of basis function this basis function may correlate with
the features in a signal is such manner, that it is impossible to distinguish these features
in the wavelet domain. Multiwavelets are a generalization of wavelets in the sense that
instead of that V 0 is spanned in the L 2 space by a basis generated (through integer shifts)
from a single scalar function ϕ(t), it is spanned by a vector (multiscaling) function ϕ(t)
[
T
[119, 76, 77, 107], which is a vector of r scaling functions ϕ(t) = ϕ [0]
(t), . . . , ϕ [r−1]
(t)]
.
The dilation equation is a vector generalization of (3.24):
ϕ(t) = √ 2
N∑
C k ϕ(2t − k), (3.48)
k=0
with H 0 (z) = C 0 + C 1 z −1 + . . . + C N z −N where each C k is an r × r coefficient matrix.
Likewise a multiwavelet function, which is a vector function ψ(t) is introduced as:
ψ(t) = √ 2
N∑
D k ϕ(2t − k). (3.49)
k=0
The entries of ψ(t) (with their integer translates) constitute a basis for W 0 with the multiwavelet
filter H 1 (z) = D 0 + D 1 z −1 + . . . + D N z −N . The Smith-Barnwell orthogonality
conditions can be imposed as:
H 0 (z)H 0 (z −1 ) T + H 0 (−z)H 0 (−z −1 ) T = 2I r , (3.50)
H 1 (z)H 1 (z −1 ) T + H 1 (−z)H 1 (−z −1 ) T = 2I r , (3.51)
H 0 (z)H 1 (z −1 ) T + H 0 (−z)H 1 (−z −1 ) T = 0, (3.52)
generalizing (3.10). The vectorfunctions ϕ(t) and ψ(t) both have compact support in
the interval [0, N] due to the assumed FIR property of the related filters. They generate
a multiresolution structure for the inner-product space L 2 (R): . . . , V −1 , V 0 , V 1 , . . ., with
⋂
l∈Z V l = {0} and ⋃ k∈Z V k = L 2 (R), analogous to the regular DWT case. Note however
that all inputs and outputs are vector sequences with r components. The input is now
recomposed of the vector sequence containing the r phases.
Multiwavelets can also be implemented with polyphase filters. The first step is to
split the input signal X(z), the low-pass filters H 0 (z) and the high-pass filters H 1 (z) into
2r phases.
X(z) = X e (z 2 ) + z −1 X o (z 2 ) (3.53)
H 0 (z) = H 0,e (z 2 ) + z −1 H 0,o (z 2 ) (3.54)
H 1 (z) = H 1,e (z 2 ) + z −1 H 1,o (z 2 ). (3.55)

3.4. MULTIWAVELETS 41
6
6
a (2,0)
z -1 x
a (1,0)
2
z -1 b (1,2) b (2,2)
z -1
a (1,1) z -1
2
a (2,1)
H p
(z)
b z
6
2
H p
(z)
b (2,0)
z -1 6
a (1,2)
2
a (2,2)
z -1 6
b (1,1)
2
b (2,1)
6
z -1
2
Figure 3.9: Multiwavelet with r = 3 in polyphase representation
Next one constructs the 2r × 2r polyphase FIR filter H p (z):
( )
H0,e (z) H
H p (z) =
0,o (z)
. (3.56)
H 1,e (z) H 1,o (z)
Then the polyphase filter H p (z) is applied, resulting in r vectors with detail coefficients
b (l,m)
k
(l being the scale and m = 0, . . . , r − 1) and r vectors with approximation coefficients
a (l,m)
k
. The approximation coefficients are then split into two phases and iterated
through the polyphase filter H p (z) as illustrated in Figure 3.9. We have:
Y k (z) =
2r−1
∑
m=0
H km (z)U m (z), (3.57)
where H km (z) is the transfer function from the m th input to the k th output. And this
can be rewritten in vector form as Y (z) = H p (z)U(z). The orthogonality conditions
translate in terms of the polyphase representation into:
H p (z)H p (z −1 ) T = I 2r , (3.58)
where I 2r is the 2r × 2r identity matrix.
Lossless systems [116, 117] or stable all-pass systems are systems that retain the
energy from the (Fourier transformable) input U to the output Y (with a possible scale

42 CHAPTER 3. WAVELET TRANSFORMATIONS
factor):
1
2π
∫ 2π
0
|Y (e iω )| 2 dω = c
2π
∫ 2π
0
|U(e iω )| 2 dω, c ∈ R + . (3.59)
It holds [117] for the transfer matrix of a stable all-pass system that:
H(z) † H(z) = cI, ∀|z| = 1, c ∈ R + , (3.60)
where † indicates the Hermitian transpose or conjugate transpose which is defined as
transposition followed by complex conjugation: (A † ) a,b = A b,a . In order to build in the
condition (3.58) the constant c is chosen to be c = 1. Note that wavelets filters are clearly
stable since they are FIR and that they are all-pass. As a result it additionally holds
that:
H(z) † H(z) ≤ I, |z| > 1, (3.61)
H(z) † H(z) ≥ I, |z| < 1. (3.62)
Similarly from the properties of lossless systems in [117]:
˜H(z)H(z) = I s , z = e iω , (3.63)
˜H(z) = H(z −1 ) T ∗ , (3.64)
where the subscript H(z) ∗ stands for conjugation of the coefficients of H(z), i.e., H(z)
has to be unitary (orthogonal for real matrices) on |z| = 1. Since we have real coefficients
in H p (z) in (3.58) and because (3.64) extends to the whole complex plane due to analytic
continuation, (3.58) is equivalent to the property of lossless systems in (3.64). Hence the
condition of orthonormality for multiwavelets comes down to requiring that H p (z) is
lossless. A real lossless polyphase matrix H p (z) of order n − 1 thus corresponds with a
pair of polynomial matrices H 0 (z) and H 1 (z) that form an orthogonal FIR (multi)wavelet
filterbank of order 2n − 1. An interpolation condition on the unit circle for the lossless
system can be used to impose a vanishing moment for the wavelets as will be further
discussed in Section 5.3. This is useful for parameterization purposes.

Chapter 4
Continuous-time analog implementation
of wavelets
In implantable medical devices such as pacemakers power consumption is a critical issue
because battery lifetime is limited. This especially holds for sensing circuits since they
are permanently active. To perform discrete-time digital signal processing, an analog
to digital (A/D) converter and sampling (continuous time to discrete time) is required
in order to transfer continuous-time analog sensor information to the digital domain.
Depending on the number of bits used, the A/D conversion is a heavy power consuming
operation. For power consumption considerations it therefore is preferable to perform as
many computations as possible in the continuous-time analog domain. In earlier work
[53, 50, 49, 54, 48] analog dynamic translinear systems (see Section 4.1) were used as a
platform to implement continuous wavelet transforms (see Section 3.1). This involved the
approximation of the wavelet transform with the impulse response of a linear system as
discussed in Section 4.2. Using the technique of Padé approximation the authors of [53]
obtained a rational approximation of the Laplace transform of the wavelet function under
approximation. In Section 4.3 this approach, that only can approximate a limited number
of wavelet functions, is discussed in more detail. A key problem of this Padé approach is
that it does not behave equally well for each time instance. An new alternative approach,
based on L 2 approximation, that can be used for a wider range of wavelet functions is
discussed in Sections 4.4 and 4.5. This approach can even be employed to approximate the
wavelet function of discrete wavelets such as the Daubechies 3 wavelet as demonstrated
in Section 4.5.
4.1 Dynamic translinear systems
The value of an input or output signal is commonly represented in circuits in the voltage
domain (volts). However Dynamic Translinear (DTL) circuits [90, 37] are current-based
43

44 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
Linear log-domain filter
Nonlinear integrator
Linear integrator
I in
Log
(Q 1 )
V in
+
-
Exp
(Q 2 )
I
1/S
(C)
V c
Exp
(Q 4 )
I out
I in
Q 1 Q 2 Q 3 Q 4
C
I0 2I0
I out
Figure 4.1: Linear log-domain lossy integrator. Note that Q 3 is not fundamental for the
operation.
circuits (amps). As a result capacitors and transistors are used in the DTL circuit,
but no resistors. The voltage-to-current transformation corresponds to an exponential
transformation [90] if, for example, bipolar transistors are used or CMOS transistors in
the region where the Voltage density is small, i.e., in the weak inversion region.
In Figure 4.1 it is illustrated how an integrator can be implemented in a DTL circuit.
It is commonly said that the integrator is implemented int the “log-domain”, referring
to the logarithmic relation between the internal state variables and capacitance voltages
and input/output voltages, since the current-to-voltage transformation corresponds to a
logarithmic transform and the voltage-to-current transformation to a exponential transform.
The nonlinear integrator is the log-domain filter in this figure [37]. The outputs
are related linearly to currents. The integrator block is nonlinear, however the overall
scheme is linearized by transforming the input/output voltages to overall input/output
currents respectively. It is well known that the for the product of two exponentials it
holds that exp a exp b = exp a + b and therefore this logarithmic relationship in DTL

4.2. WAVELET TRANSFORMATIONS AS LINEAR SYSTEMS 45
circuits makes it possible to implement multipliers. Furthermore the derivative of an
exponential function is equal to the exponential function times the derivative of the exponent.
Not only linear systems can be implemented with the DTL approach but also
some non-linear operations, such as nonlinear differential equations. Four advantages of
DTL circuits of interest are:
1. The current based scheme is potentially less power-consuming than a voltage based
scheme.
2. The absence of resistors in the physical implementation allows a smaller circuit.
3. Both linear systems as various non-linear operations (such as an RMS D/C converter,
an oscillator with limit cycle or multipliers) can be implemented with DTL
circuits.
4. An attractive potential of the DTL technique are the dynamic possibilities: The
behavior of the circuit can be changed almost instantaneously by changing the
magnitude of the currents that determine the implemented state-space system.
This can for example be used to scale the system such that a wavelet can be dilated
in order to adapt to the specific frequency that is relevant for a sense amplifier in
a given state.
In [54] a DTL approach is described which aims to implement the Gaussian wavelet
transform. This Gaussian wavelet transform is of interest for ECG processing (see e.g.
[5, 102]). The implementation is done in the analog domain for the purpose of cardiac
signal analysis. In Figure 4.2 a DTL implementation of a state-space system is displayed.
The performance of such an implementation depends largely on the accuracy of the
approximations involved in this approach. From a technological point of view, the quality
of the hardware components used in the manufacturing process may have a considerable
impact on the performance of the IC, but such issues will not be discussed here. From
a conceptual point of view, one of the critical steps concerns the approximation of the
Laplace transform of the (time-reversed and shifted) Gaussian wavelet function by means
of a strictly proper rational function of low order. For this purpose the classical technique
of Padé approximation [10, 19] was previously proposed. We will be discussing the
drawbacks of this approach and propose an alternative approach: L 2 approximation of
wavelets, that overcomes the problems discussed for the Padé approximation.
4.2 Wavelet transformations as linear systems
The available IC design methods only allow for a limited class (e.g. finite order and
causal) of linear filters to be implemented. The IC design of linear filters (Section 2.8)
of finite order is quite well understood. If a time signal f(t) is passed through a linear
system, then f(t) is convoluted with the impulse response h(t) of that linear system,
producing the output signal as in (2.26). If the continuous wavelet transform W (τ, σ)
of f(t) associated with a given mother wavelet ψ(t) on a scale σ is considered, this

46 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
I A
Integration section:
Amatrix
Vdc
+
A12
-
Wij
I
Aij (1nA
L 1nA
ij
+
Vdc A21
-
Vdc
+
A23
-
)
I A12
+
Vdc A32
-
Vdc
+
A34
-
Current matrix I Aij
(PMOS current mirros)
I A21
+
Vdc A43
-
Vdc
+
A45
-
Input section: B vector
+
Vdc
I in
+
Vdc
- B10 0 -
+
Vdc A54
-
+
Vdc A65
-
Vdc
+
-
A56
+
Vdc A76
-
Vdc
+
A67
-
+
Vdc
-
A87
Vdc
+
A78
-
+
Vdc A98
-
Vdc
+
-
A89
+
Vdc A109 -
I A910 I A1010
Vdc
+ +
Vdc
A910
- -
C 1 C 2 C 3 C 4 C 5 C 6 C 7
I B
A1010
+
C1
-
+
-C2
+
C3
-
+
C4
-
+
-C5
+
C6
-
+
-C7
+
C8
-
+
C9
-
C 8 C 9 C 10
W
j
I
Cj (1nA
)
L 1nA
j
I C1
I C2
I C10
+
Vdc
-
I out
Summation section:
C vector
I C
Current matrix I Cj
( PMOS current mirros)
Figure 4.2: Implementation of a state-space system with a DTL circuit. Picture at the
courtesy of Sandro Haddad. Note that in this specific case I C10 is negligible small [47,
p. 181].
transform is obtained as the integral defined in (3.1). Note that the wavelet transform
at a fixed scale σ involves a linear filter operation. Therefore, the analog computation
of W (τ, σ) can be achieved through the implementation of a linear filter of which the
impulse response satisfies
h(t) = 1 √ σ
ψ
( ) −t
. (4.1)
σ
For linear systems of a finite order this equation can in general not be satisfied exactly,
however a reasonably good approximation may be sufficient for the intended application.
Equation (4.1) can also be reformulated in the Laplace domain:
H(s) =
∫ ∞
0
1
√ σ
ψ
( ) −t
e −st dt. (4.2)
σ
For obvious physical reasons only the hardware implementation of strictly causal stable
filters of sufficiently low order is feasible. In other words, an implementable linear filter
will have a (strictly) proper rational transfer function H(s) that has all its poles in the
left half of the complex plane to ensure stability. In a causal system the degree of the
numerator is less than or equal to the degree of the denominator, yielding a direct feed-

4.2. WAVELET TRANSFORMATIONS AS LINEAR SYSTEMS 47
order norm shift 2.0 shift 2.5 shift 3.0 shift 3.5
3 l 1 -norm 0.997 1.466 1.871 1.980
5 l 1 -norm 0.117 0.265 0.496 0.789
7 l 1 -norm 0.017 0.027 0.071 0.153
9 l 1 -norm 0.016 0.002 0.007 0.021
3 l 2 -norm 0.397 0.551 0.678 0.696
5 l 2 -norm 0.048 0.101 0.178 0.269
7 l 2 -norm 0.007 0.010 0.025 0.053
9 l 2 -norm 0.010 0.001 0.003 0.007
3 l ∞ -norm 0.490 0.539 0.485 0.430
5 l ∞ -norm 0.077 0.151 0.232 0.303
7 l ∞ -norm 0.005 0.017 0.041 0.076
9 l ∞ -norm 0.002 0.001 0.004 0.011
3 CPU time 0.630 0.820 1.380 1.700
5 CPU time 0.860 0.990 1.240 1.430
7 CPU time 17.010 2.340 1.980 1.820
9 CPU time 933.170 62.390 10.390 5.520
energy loss 5.5 · 10 −4 7.4 · 10 −6 3.5 · 10 −8 6.1 · 10 −11
Table 4.1: Effect of time-shift on required order, norm of misfit and energy loss illustrated
on the Gaussian wavelet. The approximations were determined by using a deterministic
starting point as in Section 4.4.4. Note that the l 2 norm of the 9 th order approximation
with a time-shift of two is less well than for the 7 th order approximation. This is due to
the fact that in this case the deterministic starting point leads to a local optimum. If the
7 th order approximation was used as a starting point for the 9 th order approximation a
better result would have been obtained.
through in the case of an equality. Strict causality ensures that the direct feed-through
is absent.
Due to this strict causality h(t) will be zero for negative t, so that any time-reversed
mother wavelet ψ(−t) which does not have this property must be time-shifted, by some
value t 0 , yielding, to facilitate an accurate approximation of its (correspondingly timeshifted)
wavelet transform ˜W (τ, σ):
where
˜W (τ, σ) =
∫ ∞
−∞
x(t) ˜ψ trunc (τ − t)dt, (4.3)
˜ψ(t) = ψ(t 0 − t), (4.4)
˜ψ trunc (t) =
{
˜ψ for t ≥ 0
0 for t < 0
(4.5)
In case ψ(t 0 −t) is nonzero for t > t 0 , a truncation error results, which should be kept
small. This is illustrated in Table 4.1, where the time-reversed and time-shifted Gaussian
wavelet is approximated, using various delays. Note that an approximation error will also

48 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
occur due to the fact that a wavelet does not usually possess a rational Laplace transform
which is a requirement that follows from the restrictions to linear filters of finite order.
A possible approach for this filter synthesis problem is to use N building blocks with
each an impulse response function h k , k = 1, . . . , N and to combine them with a weight
vector w k such that h(t) = ∑ N
k=1 w kh k (t) ≈ ˜ψ trunc (t). This approach is for example
discussed in [12]. Due to the high requirements on power consumption this approach
however is not well suited for the application at hand. The system should be tailored for
the desired impulse response to keep the order and thus the power consumption as low
as possible.
4.3 Padé approximation of wavelet functions
Padé approximation provides a method to obtain a rational approximation to a given
function. In the current application a rational function in the Laplace domain is needed.
Therefore the intended application requires one to work in the frequency domain. Any
Padé approximation H(s) of ˜Ψ(s)
{ }
= L ˜ψ(t) is characterized by the property that the
coefficients of the Taylor series expansion:
˜Ψ(s) =
∞∑
k=0
1
k! ˜Ψ (k) (s 0 )(s − s 0 ) k (4.6)
of ˜Ψ(s) around a selected point s = s 0 coincide with the corresponding Taylor series
coefficients of H(s) up to the highest possible order, given the pre-specified degrees of the
numerator and denominator polynomials of H(s). If we denote the Padé approximation
H(s) at s = s 0 and of order (n, m) with n ≤ m by
H(s) = p 0(s − s 0 ) n + p 1 (s − s 0 ) n−1 + . . . + p n
(s − s 0 ) m + q 1 (s − s 0 ) m−1 + . . . + q m
(4.7)
then there are m+n+1 degrees of freedom, which generically makes it possible to match
exactly the first m + n + 1 coefficients of the Taylor series expansion of ˜Ψ(s) around
s = s 0 . In fact, this matching problem can easily be rewritten as a system of m + n + 1
linear equations in the m + n + 1 variables p 0 , p 1 , . . . , p n , q 1 , . . . , q m . See, e.g., [10].
This brings us to one of the main advantages of Padé approximation: the linear system
of equations will generically yield a unique solution which is easy to compute. Moreover,
a good match is guaranteed between the given function ˜Ψ(s) and its approximation H(s)
in a neighborhood of the selected point s 0 . However, there are also some disadvantages
which limit the practical applicability of this technique in the setting of this research.
One important issue concerns the selection of the point s 0 . Note that a good approximation
of ˜Ψ(s) over the entire (complex) Laplace domain is not a requirement per se.
Instead, an approximation is needed which performs well when used for convolution in
the time domain. Since the function ˜ψ(t) is a wavelet, it effectively will have compact
support and in particular it should be approximated well in the region of the time domain
where it ‘lives’: which is somewhere near t = 0. Now, the initial value theorem for the

4.3. PADÉ APPROXIMATION OF WAVELET FUNCTIONS 49
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
time-reversed/shifted gaussian wavelet
5th order pade s 0 =0
7th order Pade, s 0 →∞
9th order generalization Pade
−0.8
0 1 2 3 4 5 6
time−reversed/shifted gaussian wavelet
10 4 5th order pade s 0
=0
7th order Pade, s →∞ 0
10 3
9th order generalization Pade
10 2
10 1
10 0
0 1 2 3 4 5 6
Figure 4.3: Top figure: Padé approximations of the Gaussian wavelet. Lower figure: The
value of all function has been increased by 1 such they are all positive and they can be
plotted on a logarithmic scale. It is clear that the Padé approximation with s 0 → ∞ is
unstable and the Padé approximation with s 0 = 0 has a poor fit at the beginning of the
impulse response.

50 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
Laplace transformation (2.18) motivates the choice s 0 = ∞. This choice will lead to a
good approximation of ˜ψ(t) near t = 0, as is demonstrated in Figure 4.3 for a 7th order
approximation of the Gaussian wavelet.
A second important issue concerns stability. The approximation h(t) of ˜ψ(t) is required
to tend to zero for large values of t, since ˜ψ(t) has this property too. However,
stability does not automatically result from the Padé approximation technique. Indeed,
if emphasis is put on obtaining a good fit near t = 0 by choosing s 0 = ∞, it may easily
happen that the resulting approximation becomes unstable: see again the 7th order approximation
in Figure 4.3. The selection of a suitable point s 0 involves a choice between
a good fit near t = 0 and stability, yielding a non-trivial problem which may be difficult
to handle, depending on the wavelet at hand. In this respect it may be of interest to
note that the 5th order approximation described in [54] and illustrated in Figure 4.3 was
obtained with the choice s 0 = 0 which corresponds to a good fit in the time domain for
large values of t as a result of the final value theorem (2.19). This however usually results
in a poor fit at the beginning of the signal in the time domain, which is clearly visible in
Figure 4.3 too.
A third issue concerns the choice of the degrees m and n of the numerator and
denominator polynomials of the rational approximation H(s). An unfortunate choice
may yield an inconsistent system of equations or an unstable approximation. Changing
m or n may solve this problem, but the converse may also happen: one may run into
stability problems even if s 0 is left unchanged.
In [19] an overview is given of various generalizations and extensions of Padé approximation
which aim to deal with some of the problems just mentioned. For instance, it
is possible to choose some or all of the poles of the rational approximation H(s) in advance.
This offers a possibility to deal with the stability issue, but no clear theory exists
on the optimal choices for these poles. Another generalization involves the possibility
to use more than one interpolation point, which for instance offers a method to deal
with the trade-off between s 0 = ∞ and s 0 = 0 in a more systematic way. Yet another
possibility is to deal with many interpolation points s 0 , s 1 , . . . , s k and to require only a
match between the values of H(s i ) and ˜Ψ(s i ) for i = 0, 1, . . . , k, no longer taking any
derivatives at these points into account. One may even specify more interpolation points
than the number of unknowns and use a linear least squares estimation technique to
arrive at a unique solution. The advantage of such an approach is that one can optimize
the function over a better distributed and controlled set of points than with classical
Padé approximation. One may choose complex values here. A 9th order approximation
of the Gaussian wavelet function obtained with this method is illustrated in Figure 4.3.
This shows the feasibility of the approach, but for low order approximations the choice
of interpolation points becomes more critical and the results are markedly less well.
However, all of these techniques remain to have one important drawback in common:
the quality of the approximation of the wavelet is not measured directly in the time
domain but in the Laplace domain, and the criterion used does not allow for a direct
interpretation in system theoretic terms.

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 51
4.4 L2 approximation of wavelet functions
The theory of L 2 -approximation, can be formulated equally well in the time-domain and
in the frequency domain, providing an alternative framework for studying the problem
of wavelet approximation which offers a number of advantages over the Padé-approach.
The advantages of this technique will be discussed, as well as an approach to find a
suitable approximation.
4.4.1 Wavelets and the L2 space
On the conceptual level it is quite appropriate to use the L 2 -norm to measure the quality
of an approximation h(t) of the function ˜ψ(t). Indeed, the very definition of the wavelet
transform itself involves the L 2 -inner product between the signal f(t) and the mother
wavelet ψ(t). It is also desirable that the approximation h(t) of ˜ψ(t) behaves equally well
for all time instances t since h(t) is used as a convolution kernel with any arbitrary shift.
This property holds naturally for L 2 -approximation, but it is not supported by the Padé
approximation approach.
Another advantage of L 2 -approximation is that it allows for a description in the time
domain as well as in the Laplace domain, so that both frameworks can be exploited to
develop further insight. According to Parseval’s identity [80] the squared L 2 -norm of the
difference between ˜ψ(t) and h(t) can be expressed as:
‖ ˜ψ − h‖ 2 =
∫ ∞
= 1
2π
−∞
∫ ∞
(
˜ψ(t) − h(t)
) 2
dt, (4.8)
−∞
∣
∣˜Ψ(iω) − H(iω) ∣ 2 dω. (4.9)
Minimization of ‖ ˜ψ − h‖ is therefore equivalent to minimization of the L 2 -norm of the
difference between the Laplace transforms ˜Ψ trunc (s) and H(s) over the imaginary axis
s = iω. Note that this observation provides a rationale for the choice of interpolation
points in a generalized Padé-approximation approach. In addition note that due to the
causality of h(t) we can also write:
‖ ˜ψ − h‖ 2 =
∫ 0
−∞
= ζ ψ,t0 +
( ) 2
∫ ∞ ( ) 2
˜ψ(t) dt + ˜ψ(t) − h(t) dt, (4.10)
∫ ∞
0
0
(
˜ψ(t) − h(t)
) 2
dt, (4.11)
so since ζ ψ,t0 does not depend on h(t) we may restrict the criterion to the positive real
axis and the use of ˜ψ(t) in the optimization criterion is equivalent to the use of ˜ψ(t).
One of the disadvantages of an L 2 -approximation approach is that there is a risk
that the numerical optimization of ‖ ˜ψ − h‖ ends in a local, non-global optimum. Several
global optimization techniques and software packages exist and if the problem can be
rewritten in a specific form, then a global optimum can be found. However in general
there is no guarantee whether a global optimum will be found or even whether it has

52 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
been found. Different starting points can give different local optima and thus can be
used to find better solutions. Also, the outcomes of other approximation techniques can
be used as starting points for L 2 -approximation.
A question that arises is whether we can predict how good our approximation of the
wavelet transformation is; i.e. not how well the basis functions are approximated, but
how well the resulting detail coefficients match the detail coefficients that come from the
actual transformation.
Theorem 4.4.1. The squared error of the approximated wavelet transformation at any
given point at any given scale |W (τ, σ) − Ŵ (τ, σ)| is no greater than the energy E s(t)
of the signal s(t) multiplied by the squared error of the wavelet function approximation
‖ ˜ψ(t) − h(t)‖.
Proof. Denote the error of the wavelet approximation as:
Given a scaled version of the wavelet:
‖ ˜ψ(t) − h(t)‖ = ε (4.12)
ψ τ,σ = 1 √ σ
ψ
( t − τ
σ
)
, (4.13)
and the resulting wavelet transform of signal s(t) at a given scale σ and place τ:
W (τ, σ) = 〈s(t), ψ τ,σ (t)〉 =
∫ ∞
We can take their respective approximations:
( )
(
1 t − τ 1
√ ψ
∼ √ h t 0 − t − τ )
,
σ σ
σ σ
( )
1 −t + (τ + t0 σ)
= √ h
,
σ σ
−∞
s(t)ψ τ,σ (t)dt. (4.14)
= ˆψ τ,σ (t) (4.15)
〈
Ŵ (τ, σ) = s(t), ˆψ
〉
τ,σ (t) ,
=
∫ ∞
−∞
We can calculate the error of the approximation as:
|W (τ, σ) − Ŵ (τ, σ)| = | 〈s(t), ψ τ,σ(t)〉 −
s(t) ˆψ τ,σ (t)dt. (4.16)
〈
s(t), ˆψ
〉
τ,σ (t) |. (4.17)
Due to the linearity of the inner product in each of the arguments:
|W (τ, σ) −
〈s(t), Ŵ (τ, σ)| = | ψ τ,σ (t) − ˆψ
〉
τ,σ (t) |. (4.18)
Using the Cauchy-Schwarz inequality we obtain an upper bound:
√
|W (τ, σ) − Ŵ (τ, σ)| ≤ ‖s(t)‖ ‖ψ τ,σ(t) − ˆψ τ,σ (t)‖ = E s(t) ε. (4.19)

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 53
A number of remarks are in place here. First the upper bound depends on the energy
of the total signal, whereas the wavelet functions have (effective) compact support. As
a result the upper bound can be reduced by taking this into account. The upper bound
then becomes dependent on the scale and properties of the wavelet. A second point is
that the wavelets and their approximations can possess beneficial properties that further
reduce the error, among which the possession of a vanishing moment as described in
Section 4.4.3. This property avoids a bias in the approximated wavelet transform.
4.4.2 Parameterization
Particularly in the case of low-order approximation, the L 2 -approximation problem can
be approached in a simple and straightforward way in the time domain. As is well known
from linear systems theory (see, e.g., [62]) any strictly causal linear filter of finite order
n can be represented in the time domain as a state-space system (A, B, C) as in (2.28).
The impulse response function h(t) and its Laplace transform H(s) (i.e., the transfer
function of the system) are then given by:
h(t) = Ce At B, (4.20)
H(s) = C(sI − A) −1 B. (4.21)
For the generic situation of stable systems with distinct poles, the impulse response
function h(t) is a linear combination of damped exponentials and exponentially damped
harmonics. For low-order systems, this makes it possible to propose an explicitly parameterized
class of impulse response functions among which to search for a good approximation
of ˜ψ(t) as described in [67].
The type of functions that come from (4.20) are the class of Bohl functions [97,
Remark 3.5.3]. Bohl functions are sums of the products of polynomials and exponentials,
which, in the real case, comes down to the sum of the product of polynomials, real
exponentials, sines and cosines. The matrix exponential is defined as:
e At def
= I + At + A2 t 2
+ A3 t 3
+ . . . (4.22)
2 3!
The matrix A in (4.22) can be transformed, using a similarity transform, to Jordan form
(a block-diagonal matrix).
A = S −1 BS (4.23)
e At = S −1 e Bt S, B in Jordan form (4.24)
These blocks do not show interaction and each gives rise to a specific Bohl function.
Scalar blocks obviously form pure exponentials. Products of polynomials and exponentials
correspond to special Jordan structures in the matrix A and since they are not
generic, they are not used, unless one has other reasons to consider them, in the parameterization.
Typically, the functions that are used in the parameterization are damped

54 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
exponentials and exponentially damped sines and cosines. The latter can model the
oscillatory behavior of wavelets.
Sufficiently smooth continuous-time wavelets can be reasonably well approximated by
strictly causal stable linear systems with distinct poles. The selected parameterization
consists of damped exponentials and exponentially damped harmonics. The parameterized
class of impulse response functions has the following form:
h(t) = [ α 1 e p1t + . . . + α n e pnt] + [ β 1 e q1t sin(r 1 t) + γ 1 e q1t cos(r 1 t)+
. . . + β m e qmt sin(r m t) + γ m e qmt cos(r m t) ] , (4.25)
where the parameters p k and q k must be strictly negative for reasons of stability.
For instance, if a 5 th order approximation is attempted, this parameterized class of
functions h(t) may typically have the following form:
h(t) = α 1 e p1t + β 1 e q1t sin(r 1 t) + γ 1 e q1t cos(r 1 t) + β 2 e q2t sin(r 2 t) + γ 2 e q2t cos(r 2 t), (4.26)
Note that (4.26) can be rewritten as a sum of complex exponentials:
h(t) = α 1 e p1t + η 1 e (q1+ir1)t + η ∗ 1e (q1−ir1)t + η 2 e (q2+ir2)t + η ∗ 2e (q2−ir2)t , (4.27)
√ (
with p k , q k ∈ R − β 2
k
, α k , β k , γ k , r k ∈ R, η k =
+γ2 k
2
e i(arctan βk
γ
)+1 k R − (β k )π− π 2 ) and 1 A (x)
is the indicator function: 1 A (x) = 1 iff x ∈ A.
The parameterization has in this form some similarity to the classical problem of
separation of exponentials [74, Chapter IV-23]. The problem deals with a function given
in the following form:
f(x) = ρ 1 e −λ1x + ρ 2 e −λ2x + . . . + ρ m e −λmx , (4.28)
and aims to find the amplitudes ρ i and damping coefficients λ i . A difference with the
problem at hand is that complex exponentials need to be found, and as a result the
method would need modification.
For the purpose of IC design it is useful to have a state-space representation (A, B, C)
associated with (4.26) available. Such a representation is for instance provided by:
⎛
⎞ ⎛ ⎞
p 1 0 0 0 0
1
0 q 1 r 1 0 0
0
A =
0 −r
⎜ 1 q 1 0 0
, B =
1
,
⎟ ⎜ ⎟
⎝ 0 0 0 q 2 r 2 ⎠ ⎝0⎠
0 0 0 −r 2 q 2 1
C = ( α 1 β 1 γ 1 β 2 γ 2
)
. (4.29)
Given the explicit form of the wavelet ˜ψ trunc (t) and the parameterized class of functions
h(t), the L 2 -norm of the difference ˜ψ trunc (t) − h(t) can now be minimized in a
straightforward way using standard numerical optimization techniques and software. In

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 55
[67, 66] a numerical approach was used that involves discretizing both the wavelet and
the parameterized impulse response with a very fine mesh, and locally searching for the
least-squares minimum of the difference between the two.The problem of avoiding local
optima is discussed in Section 4.4.4. The negativity constraints on {p k } and {q k } which
enforce stability are not difficult to handle for most optimization packages.
4.4.3 Vanishing moments
One common property of a wavelet function ˜ψ(t) that was undiscussed so far is that it
must have at least one vanishing moment. Enforcing the first vanishing moment comes
down to the requirement that the integral of the wavelet function is equal to zero:
∫ ∞
0
˜ψ(t)dt = 0. (4.30)
If this property is not shared by the approximation h(t), this will cause an unwanted
bias in the approximation of the wavelet transform as can be seen in the simulation in
Figure 4.4, where the h(t) obtained in this manner is denoted “L 2 normal” in the figure.
This is likely to happen in a situation where a truncation error occurs.
Proposition 4.4.2. The condition ∫ ∞
0
h(t)dt = 0 is equivalent to H(0) = 0.
Proof.
0 =
∫ ∞
0
h(t)dt =
∫ ∞
0
e −st h(t)dt, with s = 0 (4.31)
= H(s) with s = 0 (4.32)
= H(0). (4.33)
In terms of linear filters, the property that the integral of the impulse response function
h(t) is zero is equivalent to the property that the step response of the filter tends
to zero for large t as follows from (2.27) and the final value theorem (2.19). Indeed, if
the wavelet transform is computed for a step input signal, then a bias will be manifest if
such a property is not satisfied.
In terms of a state-space representation (A, B, C) we have that
H(0) = −CA −1 B. (4.34)
As an example, for the representation (4.29) it is not difficult to compute A −1 since it is
block diagonal.
⎛
⎞
p 1 0 0 0 0
0 q 1 r 1 0 0
A −1 =
0 −r
⎜ 1 q 1 0 0
⎟
⎝ 0 0 0 q 2 r 2 ⎠
0 0 0 −r 2 q 2
−1
(4.35)

56 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
Simulation of various systems
0.2*Dataset
Regular wavelet
L2 normal
L2 step corr.
−1
0 50 100 150 200 250
Figure 4.4: A scaled dataset is displayed along with its transformation with a wavelet.
The black, thin, dashed line displays the output of a system that has the dataset as an
input and attempts to approximate the wavelet transform without enforcing that the
step response tends to zero for large t. The reader can observe that a bias manifests that
correlates with the input data. The diamonds show the output of another system that
does enforce the condition on the step response. For this system the bias is absent in the
output data.
⎛
A −1 =
⎜
⎝
1
p 1
0 0 0 0
(
)
0 1 q 1 −r 1
0 0
0
q1 2+r2 1 r 1 q 1 0 0
(
)
0
0 0
1 q 2 −r 2
0
0 0
q2 2+r2 2 r 2 q 2
⎞
⎟
⎠
(4.36)
This yields the explicit condition:
H(0) = α 1
+ −β 1r 1 + γ 1 q 1
p 1 q1 2 + + −β 2r 2 + γ 2 q 2
r2 1 q2 2 + = 0 (4.37)
r2 2
If such an extra nonlinear condition is not conveniently handled by the optimization
software, then it can easily be used to eliminate one of the variables from the problem:
α 1 = −p 1
(
γ1 q 1 − β 1 r 1
q 2 1 + r2 1
+ γ )
2q 2 − β 2 r 2
q2 2 + r2 2
(4.38)

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 57
Step and impulse response of L2 models with and without step correction
1
i.r. L2 normal
i.r. L2 step corr
s.r. L2 normal
0.5
s.r. L2 step corr.
0
−0.5
−1
0 5 10 15 20
Figure 4.5: Impulse and step responses of the “normal” and “step corrected” L 2 -
approximation
4.4.4 Obtaining a good starting point
When optimizing the parameterized class of impulse response functions to match a certain
wavelet function in an L 2 -sense, it is a non-trivial task to find the global optimum. There
generally exist a large number of local optima and it is very hard to find the global
optimum. In fact the problem of global optimization in this setting is unsolved and no
guarantees in general exist that a global optimum can or has been found. For specific
problems these guarantees do exist and one can attempt to rewrite problems into such a
form, however no suitable form has been found for the problem at hand. The optimization
technique used in this study is to locally minimize a discretized least-squares criterion,
since an l 2 -criterion is to be minimized. In order to prevent the optimization technique
from terminating in an unsatisfactory local optimum, one can attempt to find a good
starting point. The choice of the starting point can have a considerable impact on the
solution found by the L 2 -approximation approach.
A methodology is presented in [66] to obtain a good starting point for the L 2 -
approximation approach in an automated fashion. One starts by constructing a highorder
model and applying model reduction techniques such that an initial model with
the appropriate order n req is obtained. A number of intermediate steps are required as
illustrated in Figure 4.6 and listed below:
1. Sampling the wavelet ˜ψ

58 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
High order
discrete-time
FIR-model
Sampled
wavelet
function
Intermediate
order
discrete-time
IIR-model
L 2
approximation
Initialization
Intermediate
order
Continuoustime
model
2 3 4
Balance and
ZOH discrete to
1
Truncate
Optimization
Function
6
continuous time
Balance and
Truncate
Low order
continuoustime
model
5
Restrictions
Approximated
wavelet
function
Model class
Wavelet admissability
conditions
Figure 4.6: Automated approximation of functions
The wavelet function ˜ψ is sampled with a sufficiently high resolution over a large
enough time interval.
2. Construction of a high order discrete-time FIR-model
The sampled wavelet is used to construct a high order discrete-time FIR (or movingaverage)
model, which has an impulse response that exactly matches the sampled
wavelet.
3. Conversion to an intermediate order discrete-time IIR-model
The state-space model is balanced and truncated to yield an accurate reduced order
discrete-time model, referred to as an intermediate order model.
4. Conversion of the discrete-time IIR-model to a continuous-time IIRmodel
The discrete-time model is then converted back to continuous-time, using the ZOHprinciple.
Until here, all steps in the procedure have to be performed just once.
5. Reduction of the continuous-time IIR-model to the desired lower order
The intermediate order continuous-time model is reduced to a specified lower order
n req , to be used as a starting point for the optimization technique in the next step.
Various reduced orders can be attempted until a satisfactory result is obtained,
steps 1-4 do not have to be repeated.
6. L 2 -approximation of the wavelet function
The low order model obtained in the previous step is used as a starting point for

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 59
solving the minimization problem described above under the constraint that for
(4.34) it must hold that H(0) = 0, using an iterative local search optimization
technique.
Sampling the wavelet ˜ψ
To obtain a good starting point one can start with a sampled version of the time-reversed
and shifted wavelet ˜ψ(t) using sample intervals of size ∆t:
f k = ˜ψ(k∆t), k = 0 . . . n. (4.39)
The horizon n∆t has to be selected in such a way that stability is obtained. However
the computational complexity grows with n. In the case of non-compactly supported
wavelets the horizon can be selected by choosing a threshold α and setting n∆t close to
the smallest t max for which | ˜ψ(t)| < α, ∀t > t max . One should select α in such a way that
not only stability is obtained, but also the truncation error is sufficiently small. Stability
is not guaranteed, but implicitly obtained in practice. For compactly supported wavelets
n and ∆t should be selected in such a way that ˜ψ(n∆t) is well outside the support of
the wavelet to ensure stability, but n is small enough to keep the calculations feasible. A
typical value for ∆t is 0.01 and for n∆t a typical value is 20, depending on the decay of
the function at hand.
Construction of a high order discrete-time FIR-model
The sequence {f k } will be used as the impulse response of a discrete-time system. Caution
needs to be taken here since the impulse function in continuous-time is inherently different
from the impulse response in discrete-time. The continuous-time impulse function is the
Dirac delta-function (2.15), yielding an infinitesimally narrow, infinitely tall pulse which
integrates to unity. On the other hand the discrete-time impulse can be associated
(via zero-order hold) with a continuous time block function, called the Kronecker delta
function (2.16). As a result a correction is required by filtering the sequence {f k } by a
FIR block filter 1 2 + 1 2 z−1 to obtain { ˆf k }, which corresponds to a discrete-time impulse
response. This sequence can be then be used to define a discrete-time system.
The impulse response of the system under construction is chosen to be as follows:
h[0] = 0, h[1] = ˆf 0 , h[2] = ˆf 1 , . . . , h[n + 1] = ˆf n . The impulse response of a discrete-time
state-space system (A, B, C, d) is given by:
h[k] =
{ d k = 0
CA k−1 B k > 0.
(4.40)
Therefore the required impulse response can be obtained by using a system M dh =
(A dh , B dh , C dh , d dh ) in controllable companion form and a zero term d dh . The matrix

60 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
A dh then has the following form that shifts the state variables in time:
⎛
⎞
0 . . . 0 0
0
A dh =
⎜
⎟
⎝ I . ⎠
0
(4.41)
The input is only fed to the first state and the vector C determines the output given
the function that has to be fitted:
⎛
B dh = ⎜
⎝
1
0
.
.
0
⎞
⎟
⎠
( )
C dh = ˆf0 ˆf1 . . . ˆfn
d dh = 0
(4.42)
Eventually the model in (4.42) has to be converted to a low-order continuous time
model. It may however not be easily converted directly to a continuous time model
because A dh has all its poles at the origin, which makes it impossible to take a matrix
logarithm as can be seen from (4.57). Therefore the model M dh is first reduced, then
converted to continuous time and finally is reduced again to obtain a continuous-time
model of the required low order.
Conversion to an intermediate order discrete-time IIR-model
To reduce the model M dh the balance and truncate procedure can be used for which the
reader is referred to [89, 96, 121].
M dh has to be balanced first. A stable, discrete-time system is balanced if the following
system of equations hold where the first two equations are the well-known discretetime
Lyapunov-Stein equations:
P − AP A T = BB T (4.43)
Q − A T QA = C T C (4.44)
P = Q = diag{σ 1 , . . . , σ n } (4.45)
for positive numbers σ k , known as the Hankel singular values of the system. The system
does always admit a balanced realization and the Hankel singular values can be ordered
in a decreasing fashion. Observe that (4.43) does only depend on A and B, and as a
result not on the wavelet ˜ψ(t) that was fitted to M dh . It is not difficult to see that
the matrix P , i.e., the controllability Grammian, corresponding to the model M dh is
the identity matrix I. As a consequence P will be unaffected by orthogonal state-space
transformations.

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 61
For (4.45) to hold, the matrix Q or the observability Grammian has to be diagonalized
first by an orthogonal state-space transformation and then the system will be transformed
by an additional diagonal state-space transformation to ensure P = Q.
The diagonalization of Q is indeed possible with an orthogonal state-space transform
of the form UQU T since the matrix Q is always positive definite in case of stability
and minimality. The transform will have an influence on A, B and C , but not on P ;
an identity matrix, as mentioned earlier. This makes it possible to transform Q with
singular value decomposition.
Ā dh = U T A dh U (4.46)
¯B dh = U T B dh (4.47)
¯C dh = C dh U (4.48)
¯P = P = I (4.49)
¯Q = U T QU (4.50)
The orthogonal transformation matrix U will is such that ¯Q is a diagonal matrix. Condition
(4.45) can now be ensured by an additional diagonal state-space transformation.
The procedure discussed in [11] arrives at the same balanced realization in a slightly
different way, by observing that for the controllable companion form realization of a
moving-average system, the associated Hankel matrix H built from the finite impulse
response 0, h[1], h[2], . . . , h[n] as
⎛
⎞
h[1] h[2] h[3] . . . h[n]
h[2] h[3] h[n] 0
H =
h[3] 0 .
(4.51)
⎜
⎟
⎝ . h[n] 0
. ⎠
h[n] 0 . . . . . . 0
is diagonalized by the same matrix U. This avoids the computation of Q, of which the
condition number is the square of the condition number of H.
The controllability grammian P is defined as:
∞∑
P = A k BB T (A T ) k = CC T , (4.52)
k=0
where C = (B AB A 2 B . . .), i.e., the infinite controllability matrix. Similarly Q =
OO T , where O T = (C T A T C T (A T ) 2 C T . . .), i.e. the observability matrix. Since
h[k] = CA k−1 B and because A k−1 B = 0 for k > n, it is not necessary to consider an
infinite matrix H. The Hankel matrix can be written in terms of the observability and
controllability matrices as: HH T = OCC T O T = OO T , since P = I.
After balancing ¯Q is a diagonal matrix with the singular values {σ 1 , . . . , σ n } of Q
in decreasing order. Upon choosing a threshold ɛ σ , the state vector may be truncated,
retaining as many n m state components as there are singular values above the threshold,
yielding an n m th order discrete-time system M dm = (A dm , B dm , C dm , 0).

62 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
Conversion of the discrete-time IIR-model to a continuous-time IIR-model
The matrix A dm will usually no longer contain zero eigenvalues and therefore the system
M dm can be converted to continuous time as described in for example [115]:
A dm = e Acm∆t (4.53)
B dm =
∫ ∆t
0
e Acmτ B cm dτ (4.54)
C dm = C cm (4.55)
with A dm , A cm , B dm and B cm further conveniently related by (see [118]):
N =
e N∆t =
( )
Acm B cm
0 0
( )
Adm B dm
0 1
To compute the matrix N first diagonalize the matrix F =
(4.56)
(4.57)
( )
Adm B dm
, using that
0 1
F ′ = V −1 F V = diag(k), (4.58)
where V is the matrix with eigenvectors of F , k is the vector of eigenvalues of F and the
function diag gives a diagonal matrix the given vector as its diagonal elements, produces
a diagonal matrix F ′ if F . Next the matrix logarithm of F can be calculated as:
N∆t = ln F = V ln F ′ V −1 (4.59)
From (4.59) it can be seen that the logarithm of a matrix, of which the diagonal
elements contain the diagonal entries of the matrix A m athrmdm has to be taken. Since
the logarithm of zero is undefined one cannot transform matrices A dm that have zeros
on the diagonal to continuous-time directly. Note that the controllable companion form
with a matrix A m athrmdm as in (4.41) does have zeros on the diagonal and hence cannot
be directly transformed to continuous-time.
Reduction of the continuous-time IIR-model to the desired lower order
The model M cm will not yet have the required order n req , therefore the continuous time
model that has now been obtained has to be reduced even further with, for instance,
the balance and truncate procedure. This procedure now involves the continuous-time
Lyapunov equations, so that eventually a continuous-time model M cr of reduced order
n req is obtained.

4.4. L2 APPROXIMATION OF WAVELET FUNCTIONS 63
1
0.5
0
Morlet wavelet approximation
Morlet wavelet
5 th order approximation
7 th order approximation
8 th order approximation
-0.5
-1
0 2 4 6 8 10 12 14 16 18 20
0.4
0.2
Morlet wavelet approximation error
5 th order approximation error
7 th order approximation error
8 th order approximation error
0
-0.2
-0.4
0 2 4 6 8 10 12 14 16 18 20
Figure 4.7: Automatic approximation of the Morlet wavelet
L2-approximation of the wavelet function
M cr is a single input - single output, strictly proper, asymptotically stable, continuoustime
state-space system which will not in general be of the form as in (4.25), with a
corresponding state-space realization of a form similarly as the example in (4.29) and
will not obey the constraint as described in Section 4.4.3. To ensure that the step
response of M cr tends to zero, the constant term in the numerator of the transfer function
associated with M cr is simply set to zero to enforce a zero at zero. Next the system has
to be brought in the model form as in (4.25). The complex eigenvalues associated with
A cr will be ordered in complex pairs where the imaginary part is positive. The states
corresponding to real eigenvalues follow last. The A matrix will have a block structure,
from which the required parameters can be easily read off. Note that A cr fixes the number
of complex pairs that the approximating model will have.
In Figure 4.7 the approximation of the Morlet wavelet with systems of various reduced
orders is illustrated. In the lower half of this figure the deviation from the ideal
wavelet is shown. Since the “ideal” wavelet is truncated, it does not have a zero integral.
The approximations however, do have this property, as described in Section 4.4.3, and
therefore a perfect fit is not possible.

64 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
Abbreviation Matlab Release Function Algorithm
lsqc2008bTrr R2008b “lsqcurvefit” “Algorithm”=“trust-region-reflective”
fmin2008bTrr R2008b “fminsearch” “Algorithm”=“trust-region-reflective”
lsqc14-3GN R14 SP3 “lsqcurvefit” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
lsqc2007bGN R2007b “lsqcurvefit” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
lsqc2008bGN R2008b “lsqcurvefit” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
fmin14-3NM R14 SP3 “lsqcurvefit” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
fmin2007bNM R2007b “fminsearch” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
fmin2008bNM R2008b “fminsearch” “LargeScale”=“off”,“LevenbergMarquardt”=“off”
Table 4.2: Used Matlab settings and function for wavelet approximation. The settings
“LargeScale”=“off”,“LevenbergMarquardt”=“off” attempts to use Gauss-Newton optimization
for lsqcurvefit and Nelder-Mead simplex direct search for fminsearch. Under
certain conditions the optimization may switch to Levenberg-Marquardt or to a largescale
method. The lsqcurvefit function offers the possibility to specify upper and lower
bounds for each of the parameters.
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.0022 0.0009 0.0056 0.0070 0.0132 0.0475 0.3975
fmin2008bTrr 0.0073 0.0068 0.0066 0.0073 0.0132 0.0475 0.3975
lsqc14-3GN 0.0163 0.0198 0.0165 0.0173 0.0329 0.1168 0.9973
lsqc2007bGN 0.0044 0.0103 0.0056 0.0070 0.0132 0.0475 0.3975
lsqc2008bGN 0.0044 0.0103 0.0056 0.0070 0.0132 0.0475 0.3975
fmin14-3NM 0.0164 0.0164 0.0173 0.0199 0.0329 0.1168 0.9973
fmin2007bNM 0.0076 0.0068 0.0065 0.0077 0.0132 0.0475 0.3975
fmin2008bNM 0.0073 0.0068 0.0066 0.0073 0.0132 0.0475 0.3975
Table 4.3: Approximation of Gaussian wavelet with t 0 = 2.0 and Matlab settings as in
Table 4.2.
4.5 Empirical results
The L 2 approximation approach for wavelets allows for the implementation of a wide
variety of wavelets. In [52] it was discussed how complex wavelets can be implemented
and in [65] a large number of wavelets are approximated with this approach. Besides
the Gaussian wavelet, also the Morlet wavelet and the Mexican Hat wavelet, which
are of interest for ECG processing [63, 3, 2, 20], have been approximated. Not only
continuous wavelets have been approximated, but also discrete wavelets such as some of
the Daubechies wavelets, provided that they are sufficiently smooth. The required order
to obtain a satisfying approximation of these wavelets is quite high though.
In [65] it is shown that a 4 th order L 2 approximation of a Gaussian wavelet outperforms
a 5 th order Padé approximation, with respect to a multitude of evaluation methods,
thus showing a significant improvement in performance. The approximated wavelets are
shown in Figure 4.8.
In order to illustrate how well various wavelets are approximated the l 2 norms of the
approximation errors of various wavelets will be shown.
As can be seen from Tables 4.3–4.8, accuracy does not always increase with the
order. The use of a lower order solution as a starting point should give a performance
increase in such cases. If a local optimum is nearby, the optimization may terminate
in this local optimum. A way to overcome the latter problem is to introduce random
disturbances on the starting point and use random starting point in the area around the
deterministic starting points. Another observation is that with the same settings, the

4.5. EMPIRICAL RESULTS 65
1
(a)
1
(b)
0.5
0.5
0
0
−0.5
−0.5
−1
0 5 10 15 20
−1
0 5 10 15 20
1
(c)
1
(d)
0.5
0.5
0
0
−0.5
−0.5
0 5 10 15 20
−1
0 5 10 15 20
1
(e)
0.5
(f)
0.5
0
0
−0.5
−0.5
−1
0 5 10 15 20
−1
0 5 10 15 20
Figure 4.8: Approximations of various wavelet functions. The thick gray line represents
the wavelet function and the thin dashed black line represents the impulse response of
the approximating system. The following wavelets were approximated with the specified
orders n req and time shifts t 0 : (a) Gaussian n req = 5, t 0 = 2, (b) Morlet n req = 5, t 0 = 2.5,
(c) Mexican Hat n req = 6, t 0 = 3.2, (d) Daubechies 3 n req = 6, t 0 = −1, (e) Daubechies
7 n req = 8, t 0 = −4, (f) Coiflet 5 n req = 8, t 0 = −11.
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.0026 1.5043 0.0277 0.0636 0.0926 0.2372 0.6318
fmin2008bTrr 0.0030 0.0043 0.0277 0.0636 0.0926 0.2372 0.6398
lsqc14-3GN 0.0026 0.0037 0.0277 0.0636 0.0926 0.2372 0.6318
lsqc2007bGN 0.0026 0.0037 0.0277 0.0636 0.0926 0.2372 0.6318
lsqc2008bGN 0.0026 0.0037 0.0277 0.0636 0.0926 0.2372 0.6318
fmin14-3NM 0.0029 0.0043 0.0277 0.0636 0.0926 0.2372 0.6398
fmin2007bNM 0.0029 0.0043 0.0277 0.0636 0.0926 0.2372 0.6318
fmin2008bNM 0.0030 0.0043 0.0277 0.0636 0.0926 0.2372 0.6318
Table 4.4: Approximation of Morlet wavelet with t 0 = 2.5 and Matlab settings as in
Table 4.2.

66 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.0302 0.0051 0.0074 0.1447 0.1469 0.1782 0.9939
fmin2008bTrr 0.1077 0.1320 0.1163 0.5844 0.1720 0.1721 0.5019
lsqc14-3GN 0.0048 0.0082 0.0074 0.0189 0.0613 0.1782 0.5019
lsqc2007bGN 0.3790 1.0549 1.1359 3.0077 0.1108 1.1009 0.5019
lsqc2008bGN 0.6411 1.2933 1.0450 0.0189 0.0613 0.9974 0.5019
fmin14-3NM 0.0060 0.0052 0.0083 0.0189 0.0613 0.1782 0.5019
fmin2007bNM 0.1237 0.1871 0.2179 0.3885 0.4874 0.4850 0.5019
fmin2008bNM 0.1077 0.1320 0.1163 0.5844 0.1720 0.1721 0.5019
Table 4.5: Approximation of Mexican hat wavelet with t 0 = 3.2 and Matlab settings as
in Table 4.2.
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.8419 0.5685 0.9998 1.0001 1.0000 1.0001 1.0001
lsqc14-3GN 0.1348 0.2785 0.6065 0.6065 0.6065 0.6086 1.0001
lsqc2007bGN 0.8512 0.5835 0.8015 0.8477 1.0001 1.0001 1.0001
lsqc2008bGN 0.8419 0.5685 0.9998 1.0001 1.0000 1.0001 1.0001
fmin2007bNM ∗ 0.1440 1.0001 1.0000 0.9998 1.0001 1.0001 1.0001
fmin2008bNM ∗ 0.5749 0.5728 0.9999 1.0000 1.0000 1.0000 1.0000
Table 4.6: Approximation of Daubechies 3 wavelet with t 0 = 0.0 and Matlab settings
as in Table 4.2. The ∗ indicates that a different starting point was used to obtain the
approximation. For results close to one, the impulse response of the obtained systems
looks very much like an impulse; a very undesirable approximation.
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.0369 0.0429 0.0993 0.2430 0.2510 0.5241 0.8400
fmin2008bTrr 0.0440 0.0449 0.0993 0.2430 0.2510 0.5241 0.8400
lsqc14-3GN 0.0369 0.0429 0.0993 0.2430 0.2510 0.5241 0.8400
lsqc2007bGN 0.0369 0.0429 0.0993 0.2430 0.2510 0.5241 0.8400
lsqc2008bGN 0.0440 0.0449 0.0993 0.2430 0.2510 0.5241 0.8400
fmin14-3GN 0.0440 0.0453 0.0993 0.2430 0.2510 0.5241 0.8400
fmin2007bNM 0.0439 0.0443 0.0993 0.2430 0.2510 0.5241 0.8400
fmin2008bNM 0.0440 0.0449 0.0993 0.2430 0.2510 0.5241 0.8400
Table 4.7: Approximation of Daubechies 7 wavelet with t 0 = −4.0 and Matlab settings
as in Table 4.2.
Settings 12 th order 10 th order 8 th order 7 th order 6 th order 5 th order 3 rd order
lsqc2008bTrr 0.0282 0.0706 0.1832 0.3510 0.4340 0.6335 0.8840
fmin2008bTrr 0.0329 0.0730 0.1857 0.3510 0.0.4955 0.6335 0.8783
lsqc14-3GN 0.0284 0.0706 0.1832 0.3510 0.4340 0.6335 0.8840
lsqc2007bGN 0.0284 0.0706 0.1832 0.3510 0.4340 0.6335 0.8840
lsqc2008bGN 0.0284 0.0706 0.1832 0.3510 0.4340 0.6335 0.8840
fmin14-3GN 0.0329 0.0730 0.1857 0.3510 0.4955 0.6335 0.8783
fmin2007bGN 0.0329 0.0730 0.1894 0.3510 0.4955 0.6335 0.8783
fmin2008bGN 0.0329 0.0730 0.1857 0.3510 0.4955 0.6335 0.8783
Table 4.8: Approximation of Coiflet 5 wavelet with t 0 = −11.0 and Matlab settings as
in Table 4.2.

4.5. EMPIRICAL RESULTS 67
0.8 time−reversed, time−shifted wavelet
18th order continuous−time system
7th order system ofter model reduction
0.6
7th order system ofter local search
0.4
0.2
0
−0.2
−0.4
−0.6
0 5 10 15 20
Figure 4.9: Approximations of Mexican Hat wavelet. The time-shifted, time-reversed
Mexican Hat wavelet is displayed along its 18 th order approximation that results from
step 4 of the procedure to find a suitable starting point. The 7 th order approximation from
step 5 (which does not necessarily have an integral of 1) and the 7 th order approximation
from step 6 are displayed as well and show a poor fit with respect to the Mexican Hat
wavelet.
acquired approximation using Matlab Release 14 Service Pack 3 is markedly better than
with the newer Matlab versions.
As can be seen from Table 4.5, the 7 th order approximation of the Mexican Hat
wavelet, with the lsqcurvefit function and the Gauss-Newton algorithm, is dramatic.
This approximation is illustrated in Figure 4.9. It is clear that the starting point for
the L 2 wavelet design is very inconvenient. If one considers the Hankel singular values
corresponding to the 18 th order approximation in Figure 4.10, one can observe that
all but one states seem to contain relevant information about the system. This is
further illustrated if the corresponding impulse responses are taken into consideration in
Figure 4.11. The 17 th order system is still a good starting point and the quality quickly
decreases. How dramatic the decrease is depends on a range of factors such as the wavelet
at hand, the involved time shift, etc. And the effect of the quality of the starting point
depends on the specific optimization surface at hand, the optimization routine and even
the specific version of the software as becomes apparent from Tables 4.2–4.8.

68 CHAPTER 4. ANALOG IMPLEMENTATION OF WAVELETS
1.2
1
Hankel singular value
0.8
0.6
0.4
0.2
2 4 6 8 10 12 14 16 18
state (ordered by Hankel singular value)
Figure 4.10: Hankel singular values of the 18 th order approximation of the Mexican Hat
wavelet that results from step 4 of the procedure to find a suitable starting point.

4.5. EMPIRICAL RESULTS 69
0.8
0.6
0.4
0.2
0
−0.2
0.6
0.4
0.2
0
−0.2
0.1
0
−0.1
−0.2
−0.3
0
−0.2
−0.4
order 18
0 5 10 15 20
order 15
0 5 10 15 20
order 12
0 5 10 15 20
order 9
order 17
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0 5 10 15 20
0.6
0.4
0.2
0
−0.2
0.4
0.2
0
−0.2
0
−0.2
−0.4
order 14
0 5 10 15 20
order 11
0 5 10 15 20
order 8
0.4
0.2
0
−0.2
order 16
−0.4
0 5 10 15 20
0.2
0.1
0
−0.1
−0.2
0.2
0
−0.2
−0.4
0
−0.3347
order 13
0 5 10 15 20
order 10
0 5 10 15 20
order 7
−0.6
0 5 10 15 20
−0.6
0 5 10 15 20
−0.6695
0 5 10 15 20
Figure 4.11: Impulse responses of systems that are used as a starting point for the L 2
approximation of the Mexican Hat wavelet.

Chapter 5
Orthogonal wavelet design
An important practical issue of interest is the choice of a wavelet basis [103]. When
employing the Fourier transform the basis functions are fixed, i.e., sines and cosines.
However for wavelets more freedom exists and the basis needs to satisfy the conditions
associated with the wavelet framework. One possibility for wavelet selection is to use
a parameterized wavelet that can be tuned for the application at hand [101]. Another
possibility is to design a custom wavelet as is the case in this study [69]. Note that the
current motivation for wavelet design is not to achieve a good quality lossy compression
as is commonly seen, but feature detection, better representations and other signal
processing applications. Earlier work on the design of wavelets matched the amplitude
spectrum and the phase spectrum of a wavelet to a reference signal in the Fourier domain
separately [24]. The authors of [46] describe an algorithm for the design of biorthogonal
and semi-orthogonal wavelets. The design criterion comes down to the maximization of
the energy in the approximation coefficients, however no clear motivation is provided by
the authors. In [92] a parameterization of orthogonal wavelets, involving polyphase filters
and the lattice structure as for example in [107] (see also Section 3.2), was discussed,
however no clear design criterion was provided.
In Section 5.1 two design criteria are introduced for discrete wavelets, that can be used
to measure the quality of a discrete wavelet. This quality measure is with respect to a
given signal and the key idea behind it is the maximization of the sparsity. In Section 5.2
a parameterization of discrete-time orthogonal wavelets through polyphase filter banks
with a lattice structure as in [92, 107] is discussed. Next the enforcement of additional
vanishing moments and the use of the design criterion is discussed. Together with the
results of Chapters 3 and 4 a complete procedure for orthogonal wavelet filter design
is provided here. Using the wavelet equation (3.25) it is possible to compute a wavelet
function associated with the designed wavelet filter bank. If this wavelet function is
sufficiently smooth, it is possible to approximate the designed wavelet in analog circuits.
The steps needed to come to an approximation for the implementation in analog circuits
71

72 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
is illustrated in Figure 1.1.
When multiple well-defined features in a signal need to be distinguished simultaneously,
multiwavelets [76, 77, 107] can be employed. In the multiwavelet case multiple
bases are used to span the l 2 space. In Section 3.4 it has been discussed how these multiwavelets
can be parameterized in polyphase form and how the condition of orthogonality
comes down to requiring that system in polyphase form is “lossless” as previously discussed
in [116, 117, 107]. In order to design multiwavelets, a concrete parameterization of
lossless systems is required, which is discussed in Section 5.3. For this parameterization
in Section 5.3.1 the “tangential Schur algorithm” [55] is used. This parameterization is
first provided for scalar wavelets, followed by the parameterization for multiwavelets. In
this section the enforcement of a first vanishing moment, which is a requirement for a
valid wavelet multiresolution structure, is discussed.
5.1 Measures for the quality of a given representation
When faced with an application, the choice of a wavelet is not a trivial task. A given
wavelet can have desirable properties such as a large number of vanishing moments,
optimal ratio in the Heisenberg uncertainty rectangle, linear phase, etcetera, but this
may not give a direct answer to the question "what wavelet should be used?". In this
work a quantitative measure is described to assess the quality of a given wavelet for a
number of application types.
For compression purposes a wavelet is good for a signal if the signal can be represented
in the wavelet domain with only a few nonzero coefficients. This means that it has a sparse
representation in the wavelet domain and is well located in both time and frequency. This
is also a good property for detection purposes: if the wavelet is optimized for a certain
feature, one then can identify it at a certain scale and relate it to a specific time. If one
wants to detect anomalies, a deviation from this sparsity is a good indicator for this.
The recursive application of the wavelet filter bank to the low-pass filter outputs
(see Figure 3.3) gives a decomposition in terms of detail coefficients at all levels and in
terms of approximation coefficient(s) at the coarsest level. In case orthonormal wavelets
are employed, the energy of the signal is preserved in the detail and approximation
coefficients (using the natation in Section 3.2.3) [107, page 27]:
∑
x 2 k =
k
j∑
max
j=1
∑ (
k
b (j)
k
) 2 ∑ (
+
k
a (jmax)
k
) 2
(5.1)
Note that this is the wavelet analogue of Parseval’s identity [80] as discussed on page 4.4.1.
We introduce a vector w to consist of all these detail and approximation coefficients, i.e.
w = (b (1)
1 , . . . , b(1)
k
, b(2)
1 , . . . , b(2) k
, . . . , b(jmax) 1 , . . . , b (jmax)
k
, a (jmax)
1 , . . . , a (jmax)
k
). (5.2)
From (5.1) it follows that the l 2 -norm of this vector w (from now on one is implicitly
referred to this vector when measuring the wavelet decomposition) is not an appropriate
measure for the quality of the wavelet.

5.1. MEASURES FOR THE QUALITY OF A GIVEN REPRESENTATION 73
The guiding principle to obtain sparsity that was proposed in [69] is to aim for the
maximization of the variance [81]. This is worked out in the following two ways which
are relevant for orthogonal wavelet transforms:
1. maximization of the variance of the absolute values of the wavelet coefficients
2. maximization of the variance of the squared wavelet coefficients
The latter one maximizes the variance of the energy distribution over the detail and
approximation coefficients at the various scales, and is shown below to correspond to
maximization of the l 4 -norm. The former one corresponds to minimization of the l 1 -
norm, which is a well-known criterion to achieve sparsity in various other contexts, see
for example [33, 34, 22, 21]
Theorem 5.1.1. Let w be the vector of the wavelet and the approximation coefficients
as in (5.2), resulting from the processing of a signal x = (x 0 , x 1 , x 2 , . . . , x m ) by means of
an orthogonal filter bank. Then:
1. Maximization of the variance of the sequence of absolute values |w k | is equivalent
to minimization of the l 1 -norm V 1 = ∑ m
k=0 |w k|.
2. Maximization of the variance of the sequence of energies |w k | 2 is equivalent to
maximization of the l 4 -norm V 4 = ( ∑ m
k=0 |w k| 4 ) 1/4 .
Proof. We will prove 1. and 2. separately:
1. As expressed in (5.1), the energy E in a signal is given by E = ∑ k |x k| 2 = ∑ k |w k| 2 ,
irrespective of the choice of orthogonal wavelet basis. The variance of the vector of
absolute values {|w k |} is given by
∑k |w k| 2 (∑
k
−
|w ) 2
k|
= E (∑
m + 1 m + 1 m + 1 − k |w ) 2
k|
,
m + 1
in which E and m are constant. Hence maximization of this quantity is equivalent
to minimization of V 1 .
2. The variance of the vector of energies {|w k | 2 } is given by
∑k |w k| 4 (∑k −
|w k| 2 ) 2 ∑k
= |w k| 4
m + 1 m + 1 m + 1
( ) 2 E
−
,
m + 1
in which E and m are constant. Hence maximization of this quantity is equivalent
to maximization of the V 4 .

74 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
(a)
(b)
1350
1300
1250
1200
1150
1100
1050
1000
0.6
0.4
0.2
0
−0.2
−0.4
L 1
wavelet func
L 4
wavelet func
950
50 100 150 200 250
Time
−0.6
0 1 2 3 4 5
Time
D_6
(c)
700
600
D_6
(d)
400
D_5
500
400
D_5
300
200
Wavelet scale
D_4
D_3
D_2
300
200
100
0
−100
Wavelet scale
D_4
D_3
D_2
100
0
−100
−200
−300
D_1
50 100 150 200 250
Time
−200
−300
D_1
50 100 150 200 250
Time
−400
−500
Figure 5.1: (a) Smoothed ECG beat of 256 samples. (b) Designed wavelets for ECG
beat with filter length 8. (c) Wavelet decomposition of ECG beat with l 1 -optimized
wavelet. In the top row the coarsest detail coefficients are displayed. The intensity of the
blocks corresponds to the coefficient values. The lower rows show finer detail coefficients.
The approximation coefficients have been omitted since their large values may hamper
the readability of the detail coefficients. (d) Wavelet decomposition of ECG beat with
l 4 -optimized wavelet.
Example 5.1.2. As an example, take the smoothed ECG beat that is displayed in Figure
5.1 along with the wavelets with filter length 8 that have been designed for this signal
by optimizing the criteria V 1 and V 4 , respectively. In the lower part of the figure the
wavelet decomposition of the signal using the respective designed wavelets is displayed.
From Figure 5.1c and 5.1d it can be clearly seen that just a few wavelet coefficients are
significantly larger than the rest as expected.
5.2 Wavelet parameterization and design
For wavelet design, it is inconvenient to take the set of filter coefficients directly as a
search space for numerical optimization, on which additional constraints are imposed

5.2. WAVELET PARAMETERIZATION AND DESIGN 75
(a)
(b)
a
b
c
d
+
+ z -1
Figure 5.2: Two examples of lattices
to meet all of the necessary and desired conditions. Instead we aim to construct a
parameterization which builds the desired properties in. In this research the choice
is made to restrict to orthogonal wavelets. One of the advantages of this choice is
that the inverse transform will be the adjoint of the forward transform H −1 = H † .
Using the polyphase representation (Section 3.2.7) offers the advantage that the analysis
and synthesis filters are adjacent, without down- and upsampling in between, making
is easier to build in the required conditions for orthogonal wavelets from Section 3.2:
normalization, double shift orthogonality and a first vanishing moment. The lattice
structure [107] or all-pass systems are used to ensure that the filter is orthogonal as
discussed below.
5.2.1 Lattice structure
If one has a orthonormal polyphase matrix H p (z), it can be expressed in lattice form.
A lattice structure implements a filter H p (z) as a cascade of simple building blocks that
each implement a single multiplication. In lattice form, wavelets can be implemented as a
cascade of constant matrices and delays (see for example [107, section 4.5] and [116]). In
Figure 5.2 two examples of lattices are displayed. The lattice in Figure 5.2a corresponds
to the following polyphase filter matrix:
[ ] a b
. (5.3)
c d
And the lattice in Figure 5.2b with the matrix as in (5.6).
As noted in for example [107] in order to have symmetry and consequently linear
phase filters one can choose a = d and b = c. For orthonormal filters one chooses a = d
and b = −c and, if desired, one appropriately normalizes the coefficients. The lattice
structure is convenient since a cascade of linear phase filters is again linear phase and a
cascade of orthogonal filters is again orthogonal. Depending on the design, each lattice
will be constructed to be linear phase or orthogonal; quantization errors with respect to

76 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
2
+
cos 1
z -1 -sin 1
sin 1
2
+
cos 1
...
...
z -1
+
cos n
-sin n
sin n
+
cos n
-1
Figure 5.3: Lattice structure for a polyphase filter bank
the parameters will not change this. Any 2-channel orthonormal filter can be expressed
in lattice form [107].
When choosing a = d and b = −c and using a suitable normalization, the lattice
structure can also be implemented with a rotation matrix:
[ ]
cos θ − sin θ
R =
. (5.4)
sin θ cos θ
With this structure the 2n low pass filter coefficients ( as in (3.6)) c 0 , c 1 , . . . , c 2n−1
can be reparameterized in terms of n new parameters θ 1 , θ 2 , . . . , θ n . Note that, unlike in
[107], conventional rotation matrices are used.
For k = 1, . . . , n, let
[ ]
cos θk − sin θ
R(θ k ) =
k
, (5.5)
sin θ k cos θ k
and let
Λ(z) =
Then consider the 2 × 2 matrix product
[ ] 1 0
0 z −1 . (5.6)
H(z) = R(θ n )Λ(z)R(θ n−1 )Λ(z) · · · R(θ 2 )Λ(z)R(θ 1 )Λ(−1). (5.7)
The polyphase filter structure that follows from (5.7) is illustrated in Figure 5.3. With
this lattice structure the orthogonality constraints from Section 3.2.2 are automatically
satisfied [107].
In Section 3 the convention is used that the energy of the wavelet low- and highpass
filter coefficients is normalized to one:
∑
k c2 k = ∑ k d2 k
= 1. A filter H(z) =
( )
∑ n−1 c2k c 2k+1
k=0
z −k that forms a wavelet filter bank will now be constructed. The
d 2k d 2k+1
coefficients c k and d k should therefore obey the conditions discussed in Section 3 (Normalization
and double shift orthogonality and a vanishing moment). The coefficients
d k will be related to the coefficients c k according to the alternating flip construction
d k = (−1) k c 2n−1−k . For the sum of each phase (even and odd) of the filters the following
holds [107]:

5.2. WAVELET PARAMETERIZATION AND DESIGN 77
Proposition 5.2.1. For relation between 1) the sum of the even phase of the low-pass
coefficients c 2(l−1) , 2) the sum of the odd phase of the low-pass coefficients c 2l−1 , 3)
the sum of the even phase of the high-pass coefficients d 2(l−1) and 4) the sum of the
odd phase of the high-pass coefficients d 2l−1 respectively, and the parameters θ k of the
polyphase representation of wavelets in lattice form the following holds:
(
n∑
n
)
∑
c 2(l−1) = cos θ k (5.8)
l=1
k=1
(
n∑
n
)
∑
c 2l−1 = sin θ k
l=1
k=1
(
n∑
n
)
∑
d 2(l−1) = − sin θ k
l=1
k=1
(5.9)
(5.10)
(
n∑
n
)
∑
d 2l−1 = cos θ k . (5.11)
l=1
k=1
Proof. We shall prove this proposition by induction. For n = 1 the wavelet filter coefficients
become: c 0 = cos θ 1 , c 1 = sin θ 1 , d 0 = − sin θ 1 and d 1 = cos θ 1 for which the
theorem clearly holds. Now assume that for n = p the proposition holds, making the
∑ p
following equations true:
l=1 c 2(l−1) = cos ( ∑ p
k=1 θ k), ∑ p
l=1 c 2l−1 = sin ( ∑ p
∑ k=1 θ k),
p
l=1 d 2(l−1) = − sin ( ∑ p
k=1 θ k) and ∑ p
l=1 d 2l−1 = cos ( ∑ p
k=1 θ k). Now consider Figure
5.3. From the lattice structure it can be seen that for n = p + 1 it holds for the even
phase that:
(
n∑
p∑
)
( p∑
)
c 2(l−1) = cos θ p+1 cos θ k − sin θ p+1 sin θ k
k=1 k=1
l=1
= cos
( ∑p+1
)
θ k .
k=1
For the odd phase it holds that:
(
n∑
p∑
)
( p∑
)
c 2l−1 = sin θ p+1 cos θ k + cos θ p+1 sin θ k
k=1 k=1
l=1
= sin
( ∑p+1
)
θ k .
k=1
For the even phase of the high-pass filter it holds that
(
n∑
p∑
)
( p∑
)
d 2(l−1) = − sin θ p+1 cos θ k − cos θ p+1 sin θ k
k=1 k=1
l=1
= − sin
( ∑p+1
)
θ k .
k=1

78 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
For the odd phase it holds that:
(
n∑
p∑
)
( p∑
)
d 2l−1 = − sin θ p+1 sin θ k + cos θ p+1 cos θ k
k=1 k=1
l=1
= cos
( ∑p+1
)
θ k .
k=1
which completes the proof.
Due to (5.8) and (5.9) the identity
( ∑
l
c 2(l−1)
) 2
+
c 2l−1
) 2
= 1 (5.12)
( ∑
l
holds.
From Figure 5.3, the relation between the parameters θ k and the filter coefficients
can be observed. If the filter coefficients for a lattice structure with n = p are known,
then their relation with the filter coefficients for the case n = p + 1 and the additionally
involved parameter θ p+1 can be easily found.
Observation 5.2.2. Suppose that the p × 2 array with filter coefficients for the case
n = p is known to be
⎛
⎞
c 0 c 1
c 2 c 3
⎜
⎟
⎝ . . ⎠ .
c 2p−1
c 2(p−1)
Then the (p + 1) × 2 array with filter coefficients for the case n = p + 1 is:
⎛
⎞
c 0 cos θ p+1 c 0 sin θ p+1
c 2 cos θ p+1 − c 1 sin θ p+1 c 2 sin θ p+1 + c 1 cos θ p+1
c 4 cos θ p+1 − c 3 sin θ p+1 c 4 sin θ p+1 + c 3 cos θ p+1
.
.
⎜
⎟
⎝c 2(p−1) cos θ p+1 − c 2p−3 sin θ p+1 c 2(p−1) sin θ p+1 + c 2p−3 cos θ p+1 ⎠
−c 2p−1 sin θ p+1 c 2p−1 cos θ p+1
as follows from the lattice structure in Figure 5.2.
⎛
=
⎜
⎝
⎞
c 0 0
c 2 c 1
c 4 c 3
R(θ p+1 ) T (5.13)
. .
⎟
c 2p−3 ⎠
0 c 2p−1
c 2(p−1)

5.2. WAVELET PARAMETERIZATION AND DESIGN 79
Example 5.2.3. As an example the low-pass coefficients for the filters of orders n = 2 and
n = 3 are given:
n = 2 c 0 = cos θ 1 cos θ 2
c 1 = cos θ 1 sin θ 2
c 2 = − sin θ 1 sin θ 2
c 3 = sin θ 1 cos θ 2
n = 3 c 0 = cos θ 1 cos θ 2 cos θ 3
c 1 = cos θ 1 cos θ 2 sin θ 3
c 2 = − sin θ 1 sin θ 2 cos θ 3 − cos θ 1 sin θ 2 sin θ 3
c 3 = − sin θ 1 sin θ 2 sin θ 3 + cos θ 1 sin θ 2 cos θ 3
c 4 = − sin θ 1 cos θ 2 sin θ 3
c 5 = sin θ 1 cos θ 2 cos θ 3
In order to avoid a bias in the wavelet transform and in order to make it possible
to obtain a well-defined scaling and wavelet function (see Section 3.2.4 and [107, section
6.1]), the wavelet filter must possess at least one vanishing moment. As explained in
Section 3.2.5, when a wavelet has vanishing moments, this offers a number of advantages
such as giving the wavelet regularity and smoothness. In terms of the filter coefficients
this comes down to the constraint:
∑
c 2(l−1) − ∑ c 2l−1 = 0. (5.14)
l
l
In terms of the parameters θ 1 , . . . , θ n this comes down to the following:
Theorem 5.2.4. Consider a polyphase filter in lattice structure with the parameters
θ 1 , . . . , θ n . For this filter to have at least a single vanishing moment the condition is
[107, Theorem 4.6]:
n∑
θ k = π + l2π, l ∈ Z (5.15)
4
k=1
Proof. In terms of the filter coefficients the condition comes down to ∑ n
l=1 c 2(l−1) =
∑ n
l=1 c 2l−1. From Proposition 5.2.1 it follows that this is equivalent with the condition:
cos ( ∑ n
k=1 θ k) = sin ( ∑ n
k=1 θ k). This condition is obviously satisfied if ∑ n
k=1 θ k = π 4 +
lπ, l ∈ Z. However, when using the sign convention that ∑ 2n−1
k=1 c k = √ 2, it is required
that cos ( ∑ n
k=1 θ k) = sin ( ∑ n
k=1 θ k) = 2√ 1 2, which excludes half of the previous solutions.
∑ n
Therefore the constraint becomes:
k=1 θ k = π 4 + l2π, l ∈ Z, which completes the
proof.
5.2.2 Enforcing additional vanishing moments
To impose additional vanishing moments is possible, but substantially more complicated
since the parameters θ 1 , . . . , θ n enter the equations in a nonlinear way. As an example
the condition to have a second vanishing moment is now worked out.

80 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Theorem 5.2.5. In order for the wavelet low-pass filter resulting from the lattice structure
to have two vanishing moments, for the parameters θ 1 , . . . , θ n it must hold in addition
to (5.15) that
( ( ))
n−1
∑
k∑
cos 2 θ l + 1 2 = 0 (5.16)
k=1
l=1
Proof. The condition of the first vanishing moment H 1 (1) = 0 with the corresponding
condition on the parameters {θ k } in (5.15) is assume to hold. For the second vanishing
( ) ( )
1
moment the additional condition H 1(1) ′ = 0 must hold. Since H(z 2 H0 (z)
)
z −1 = ,
H 1 (z)
we have that: ( )
( ) ( )
d H0 (z)
1
0
= 2zH ′ (z 2 )
dz H 1 (z)
z −1 + H(z 2 )
−z −2 . (5.17)
From (5.7) we find:
( 1
√ )
H(1) =
2 2
1
2√
2
√ √
1
2 2 −
1
, (5.18)
2 2
so that the last term of (5.17) at z = 1 yields:
( ( )
0 −
1
H(1) = 2√
2
√ . (5.19)
−1)
2
Now we determine H ′ (z):
H ′ (z) = R(θ n )Λ ′ (z)R(θ n−1 )Λ(z)R(θ n−2 )Λ(z) . . . Λ(z)R(θ 1 )Λ(−1)
+ R(θ n )Λ(z)R(θ n−1 )Λ ′ (z)R(θ n−2 )Λ(z) . . . Λ(z)R(θ 1 )Λ(−1)
+ R(θ n )Λ(z)R(θ n−1 )Λ(z)R(θ n−2 )Λ ′ (z) . . . Λ(z)R(θ 1 )Λ(−1)
+ . . .
1
2
+ R(θ n )Λ(z)R(θ n−1 )Λ(z)R(θ n−2 )Λ(z) . . . Λ ′ (z)R(θ 1 )Λ(−1) (5.20)
Since we are interested in H ′ (z 2 ) at z = 1 we note that Λ(1) = I, Λ(z) becomes identity
( ) 0 0
and Λ ′ (1) = . The following is obtained:
0 −1
n−1
∑
(( (∑ n cos
H ′ (1) =
l=k+1 θ l)
sin (∑ n
k=1
l=k+1 θ )
l
⎛
)
⎛
n−1
∑
= ⎝
k=1
(
⎝ cos ∑k
l=1 θ l
( ∑k
)
sin
l=1 θ l
sin
(∑ n
l=k+1 θ l)
sin
( ∑k
l=1 θ l
− cos (∑ n
l=k+1 θ l)
sin
( ∑k
l=1 θ l
− sin (∑ n
l=k+1 θ ) ) ( )
l 0 0
cos (∑ n
l=k+1 θ )
l 0 −1
( ∑k
) ⎞
− sin
l=1 θ ( ) ⎞
l
( ∑k
) ⎠ 1 0
⎠
cos
l=1 θ 0 −1
l
)
− sin (∑ (
n
l=k+1 θ ∑k
)
l)
cos
l=1 θ l
)
cos (∑ (
n
l=k+1 θ ∑k
)
l)
cos
l=1 θ l
⎞
⎠ .
(5.21)

5.2. WAVELET PARAMETERIZATION AND DESIGN 81
( 1
Now H ′ (1) is determined:
1)
( 1
H ′ (1)
1)
=
=
=
=
=
⎛
n−1
∑
⎝ sin (∑ n
l=k+1 θ l) ( ( ∑k
) (
sin
l=1 θ ∑k
)) ⎞
l − cos
l=1 θ l
k=1
cos (∑ n
l=k+1 θ l) ( ( ∑k
) (
cos
l=1 θ ∑k
)) ⎠
l − sin
l=1 θ ,
l
⎛√ (∑
n−1
∑
n 2 sin
l=k+1
⎝
θ ) ( ∑k
) ⎞
l sin
l=1 θ l − π 4
√ (∑ (
n
k=1 2 cos
l=k+1 θ l)
sin
π
4 − ∑ ) ⎠
k
l=1 θ ,
l
⎛
n−1
∑ − √ (
2 sin 2 π
⎝
4 − ∑ ) ⎞
k
l=1 θ l
√ (
k=1 2 cos
π
4 − ∑ ) (
k
l=1 θ π
l sin
4 − ∑ ) ⎠
k
l=1 θ ,
l
⎛ (
n−1
∑
⎝ −√ 2 sin 2 π
4 − ∑ ) ⎞
k
l=1 θ l
√ (
1
k=1 2 2 sin
π
2 − 2 ∑ ) ⎠
k
l=1 θ ,
l
⎛ (
n−1
∑
⎝ −√ 2 sin 2 π
4 − ∑ ) ⎞
k
l=1 θ l
√
2 cos
(2 ∑ ) ⎠
k
l=1 θ . (5.22)
l
k=1
1
2
Due to Theorem 5.2.4 it also holds that:
( 1
H ′ (1) =
1)
n−1
∑
k=1
( 1
√ ( ∑ n
2 2 cos 2
l=k+1 θ l
( ∑
1
n
2√
2 sin 2
l=k+1 θ l
) ) −
1
2√
2
) . (5.23)
Since the condition for the second vanishing moment is D ′ (1) = 0 the bottom row of
(5.17) needs to be equal to zero. From (5.17), (5.19) and (5.22) we find that the condition
for the second vanishing moment comes down to:
which completes the proof.
( (
n−1
∑
cos 2
k=1
))
k∑
θ l + 1 = 0, (5.24)
2
l=1
The reader should note that this condition cannot be conveniently enforced by eliminating
a parameter. As a concrete example, consider the case with three parameters
(n = 3) of which θ 1 is fixed in order to enforce a first vanishing moment. In this case the
condition (5.16) becomes:
cos (2θ 2 + 2θ 3 ) + cos (2θ 3 ) + 1 = 0. (5.25)
2
Given that the condition for the first vanishing moment is satisfied it may happen that
it is not possible to satisfy the condition for the second vanishing moment by fixing
another parameter. For n = 3, the curve implicitly parameterized by (5.24) is displayed
in Figure 5.4.

82 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Figure 5.4: Curve where (5.25) is satisfied. The dot marks the location of the
Daubechies 3 wavelet in the parameter space.
5.2.3 Design and optimization
The parameterized lattice structure discussed previously allows for the design of wavelets,
where orthogonality is built into the model class. The parameters θ 1 , . . . , θ n can be chosen
to optimize a certain design goal as discussed in Section 5.1. The issues involved with
optimizing over the previously discussed parameter space are discussed in this section.
Local optima
With the criteria described in Section 5.1, combined with the theory of polyphase filters
and the lattice structure, it is possible to construct an orthogonal wavelet that is optimal
in the sense of a chosen criterion function. To compute this orthogonal wavelet,
a local search method can be employed to search the parameter space θ 1 , . . . , θ n for an
optimal wavelet. As often happens with local search techniques, there is a risk that the
optimization may terminate in a local optimum, as will be illustrated in the following
example.
Example 5.2.6. The smoothed ECG signal from Figure 5.1a is used as a prototype signal.
The number of parameters is chosen to be n = 3, yielding two degrees of freedom and a
resulting filter of length 6. As design criterion the l 1 -norm of the wavelet coefficients of
the wavelet transform is minimized. A single vanishing moment was enforced. During
the experimentation we discovered a number of local minima, as illustrated in Figures
5.5, 5.6 and 5.7.

5.2. WAVELET PARAMETERIZATION AND DESIGN 83
−1.5
−1
−0.5
0
0.5
1
1.5
−1.5
−1
x 10 4 θ 3
2
Criterion value
1.5
1
θ 2
−0.5
0
0.5
1
1.5
Figure 5.5: The existence of local minima
In Figure 5.6 and Figure 5.7 the local optima are further investigated in detail. In the
upper left corner of the figure the location of the local optima is displayed in a contour
plot of the l 1 -criterion over the free parameter space. In the other subfigures the wavelet
and scaling function corresponding to the optima in the contour plot are displayed. From
this figure it can be seen that some of these local optima can be quite similar, where one
is a flipped or shifted version of another one, yielding almost the same result. Note that
local optimum 3 is so close to number 8 that their difference is negligible. It is contained
in the same gully due to the periodicity of the parameter space. The fact that this
local optimum is found from both sides of the parameter space substantiates that this
is indeed a local optimum and not a border artefact. If one considers the low-pass filter
[−0.0016, −0.1225z −1 , 0.2217z −2 , 0.8359z −3 , 0.4870z −4 , −0.0062z −5 ] and the high-pass
filter [−0.0062, −0.4870z −1 , 0.8359z −2 , −0.2217z −3 , −0.1225z −4 , 0.0016z −5 ], associated
with this local optimum, it is easy to see that these effectively have a filter of length four,
since leading or trailing coefficients are not significantly large.
Observation 5.2.7. An interesting consequence of the preceding example is that if one
compares the significant filter coefficients of local optima 3 and 8 with the low-pass filter
of the Daubechies 2 wavelet: [−0.1294, 0.2241z −1 , 0.8365z −2 , 0.4830z −3 ] and high-pass

84 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
2
1
1
Ψ
φ
2
1
Ψ
φ
2
0
0
−1
0 1 2 3 4 5
2
1
0
Ψ
φ
−1
0 1 2 3 4 5
2
0
φ
−2
0 1 2 3 4 5
2
1
0
Ψ
φ
−1
0 1 2 3 4 5
3
5
7
Ψ
−1
0 1 2 3 4 5
2
1
0
−1
0 1 2 3 4 5
2
1
0
−1
0 1 2 3 4 5
2
1
0
Ψ
φ
−1
0 1 2 3 4 5
4
6
8
Ψ
φ
Ψ
φ
Figure 5.6: Local optima in detail. The wavelet and scaling functions corresponding to
the local optima found in the design of a wavelet for the signal in Figure 5.1a using three
free parameters.

5.2. WAVELET PARAMETERIZATION AND DESIGN 85
1.5
1
4
6
8
7
0.5
θ 3
0
−0.5
−1
1
2
−1.5
5
3
−1.5 −1 −0.5 0 0.5 1 1.5
θ 2
Figure 5.7: The location of the local optima with respect to the free parameters θ 2 and
θ 3 as in Figure 5.6 is displayed.

86 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Start
Generate
random
θ 2 ,…,θ n
Set θ 1 such
that
∑θk= π/4+ k2π
Determine H 0 (z)
and H 1 (z) from
θ 1 ,…,θ n
Determine
improved
θ 2 ,…,θ n
Decompose
the signal x
with filters
H 0 (z) and H 1 (z)
Maximum
iterations
reached?
Solution
converged?
Determine criterion
value V 1 or V 4
Figure 5.8: Algorithm to find an optimal wavelet
filter of the Daubechies 2 wavelet: [−0.4830, 0.8365z −1 , −0.2241z −2 , −0.1294z −3 ], they
are nearly identical. This observation provides a rationale for the use of the Daubechies
2 wavelet for ECG processing.
It can be concluded that local optima are present in the search space associated with
the parameterization described in this section and the criteria described in Section 5.1.
The problem of finding a global optimum in the presence of local optima is very hard.
There are a number of ways in which these local optima can be avoided. There are a
number of techniques that attempt to find a global optimum in the presence of local
optima such as for example simulated annealing [71], genetic algorithms [42] and branch
and bound methods [6]. However in general these methods cannot guarantee that a
global optimum can be found and even that an optimum that has been found is indeed
a global or local optimum. In light of these drawbacks and the applications at hand it
was decided to run the local search algorithm multiple times with a number of different,
randomly selected starting points as a heuristic of finding a global optimum.
Optimization algorithm
As discussed in the previous section a local search technique was employed and this search
algorithm was used a number of times with random starting points. This approach is
illustrated in Figure 5.8, where the blocks with solid borders visualize the local search
method and the blocks with the dashed borders represent the loop with random starting
points. This approach is implemented in Matlab, using the implementation of the Nelder-
Mead direct search (simplex) algorithm [73] that is available in the function fminsearch

5.2. WAVELET PARAMETERIZATION AND DESIGN 87
of the Matlab Optimization Toolbox.
5.2.4 Experimentation
In order to verify the effectiveness of the proposed approach a number of tests were
conducted.
Test for reproducibility
In order to validate the approach, first consider that if a signal has a sparse representation
in the wavelet domain (when transformed with a certain wavelet), this wavelet is
likely to give a good performance for the criteria discussed earlier with respect to the
wavelet decomposition of the signal in question. Note that although both the criteria V 1
and V 4 aim at maximizing the variance, they still may give different results. However
when starting with a sparse representation in the wavelet domain and reconstructing an
artificial signal, the chance that differently shaped wavelets perform equally well becomes
smaller. As a validation of the approach, we start with a sparse set of wavelet coefficients
and wavelet filter. The associated time series is used as an input for the wavelet design
procedure, and it is verified whether the wavelet filter that is discovered as and optimum
is equal to the wavelet filter used to reconstruct the signal from the wavelet coefficients.
As example of the test, first random parameters θ 2 and θ 3 are selected and θ 1 is
fixed such that (5.15) holds and the corresponding sequences of scaling and wavelet filter
coefficients (resp. H 0 and H 1 ) are determined. Next the full wavelet decomposition of a
signal of length 256 is considered and only a few of these wavelet coefficients are assigned
non-zero values. Carrying out the synthesis algorithm, we are constructing a signal in the
wavelet domain. These wavelet coefficients are considered to be a sparse representation
of an unknown signal x in terms of H 0 and H 1 . Now the signal x is determined by taking
the sparse wavelet decomposition and performing an inverse wavelet transform on it with
orthogonal wavelet filters H 0 and H 1 , i.e. the signal x is reconstructed from the sparse
wavelet representation using the wavelet filters H 0 and H 1 .
We now have a signal x for which we know that H 0 and H 1 will give a sparse representation,
thus this wavelet filter bank should perform well with respect to the optimization
criterion and there should be a reasonably large chance that this wavelet filter bank is
found as an optimum. To verify this, the signal x was taken and the optimal wavelet was
determined with the approach described in this report. Optionally, before determining
the optimal wavelet, additive white Gaussian noise was added to the signal x in order to
test the effect of noise on the approach. The testing procedure is visualized in Figure 5.9
and the results for 250 trials are displayed in Table 5.1. If the l 1 -norm of the difference
of the six actual and calculated filter coefficients is larger than 0.01, then the test is
considered to be a failure.
Without noise the original wavelet was recovered in virtually all cases as can be seen
in Table 5.1. If the SNR decreases a clear sparse representation of the signal in the
wavelet domain may no longer exist but, in particular for the l 4 criterion, the original

88 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Select random
parametersθ 1
, θ 2
, θ 3
Determine wavelet
filter from parameters
Compare
wavelet filters
Select small number of
nonzero wavelet coefficients
Reconstruct signal
in time domain with
inverse wavelet
transform
Find optimal
wavlet with
n=3 for given
signal
1
1
Optionally: Add white noise
Figure 5.9: Test for reproducibility
SNR l 1 recovery rate l 4 recovery rate
∞ 99.2% 100%
40 dB 99.6% 99.6%
20 dB 32.4% 44.4%
10 dB 3.60% 9.60%
0 dB 0.00% 1.60%
Table 5.1: Percentage of times that a wavelet that provided a sparse representation of
the prototype signal was recovered. Noise was added to the prototype signal yielding a
Signal to Noise Ratio as indicated in the table.
wavelet was still found as an optimum in many cases. Taking the presence of local optima
in consideration the results are fair.
Filter length versus criterion value
The filter length of the optimized wavelet is twice the number of parameters n. A large n
gives more freedom to optimize the wavelet with respect to the criterion since in order to
enforce the required first vanishing moment, a single parameter has to be fixed. However,
as n increases, the complexity of the optimization increases as well, and it becomes more
difficult to avoid local optima. The “curse of dimensionality” applies here: choosing
random initial points to cover the domain becomes more and more cumbersome; there
is an exponential increase of effort. To determine the effect of the choice of n on the
criterion value that is achieved during the optimization a test was conducted. For the
data in Figure 5.1 the optimal wavelet was determined for n = 1, 2, . . . , 25. For each
choice of the number of parameters, i.e. for each choice of the filter length, a thousand
random starting points were generated and the best criterion value in terms of both the

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 89
(a) − l 1
criterion
(b) − l 4
criterion
10500
−1800
10000
−1850
9500
−1900
fval
9000
fval
−1950
8500
8000
7500
−2000
−2050
5 10 15 20 25
Number of parameters
5 10 15 20 25
Number of parameters
Figure 5.10: criterion value vs n for the l 1 criterion in (a) and for the l 4 criterion in
(b). Note that the values of the l 4 criterion have been multiplied with −1 such that the
optimization problem becomes a minimization problem.
l 1 and l 4 criterion were stored. The results are illustrated in Figure 5.10. From this figure
it can be concluded that it is not beneficial for the given signal and criterion to use more
than eight parameters. The increase in criterion value for a large number parameters is
a consequence of the existence of local optima.
5.3 Multiwavelet parameterization and design
If one wants to discriminate between two features in a signal with a different morphology
then multiwavelets (Section 3.4) [119, 116, 76, 77, 107] are a powerful tool. As discussed
in Section 3.4 multiwavelets are a number of mutually orthogonal wavelet functions. In
that section it was discussed that for orthogonal multiwavelets with compact support in
a polyphase representation the polyphase matrices become lossless [116, 117] and FIR.
These orthogonal multiwavelets can be designed using a parameterization as lossless systems,
by jointly optimizing each multiwavelet function with respect to different segments
of a prototype signal as first discussed in [95]. The class of orthogonal multiwavelets with
compact support is parameterized with the results from [55]. The same design criteria
as discussed in Section 5.1 are used for multiwavelet design.
First, in the following subsection, the parameterization for (multi)wavelets using lossless
systems will be discussed. This employs interpolation theory and the theory of balanced
realizations of lossless systems by means of the tangential Schur algorithm [55].
The approach consists of a number of steps. A key observation to start from is that
a balanced realization of a lossless system has a unitary realization matrix. One way
of parameterizing unitary realization matrices is by building them as a product of elementary
unitary matrices. For this purpose, mappings F U,V are introduced. These

90 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
mappings are a special simple kind of so called “linear fractional transformations”, which
are extensively used in interpolation theory. The tangential Schur algorithm allows one
to parameterize lossless transfer functions through a recursive procedure which involves
linear fractional transformations; this procedure was cast in state-space terms in [55]. To
parameterize real FIR lossless polyphase matrices, a number of special choices for the
parameters in the tangential Schur algorithm can be made, which substantially simplify
the expressions.
Then, in the next subsection, the use of the tangential Schur algorithm [55] to parameterize
scalar wavelets is explained. It is shown how to recover the product formula
(5.7) encountered in the lattice filter implementation by making suitable choices in the
algorithm. The current set-up allows us to generalize this to the case of orthogonal multiwavelets
with compact support. We also manage to build in a first “balanced vanishing
moment”, which is much needed for practical purposes. This balanced vanishing moment
was previously enforced in the literature as a constraint, but in this work a parameterization
for multiwavelets with a built-in balanced vanishing moment is presented. In the
last subsection a design approach for multiwavelets is discussed in which the multiwavelet
parameterization is used to arrive at concrete results.
5.3.1 Parameterization of lossless systems
As discussed in Section 3.4 the condition of orthogonality for polyphase multiwavelet
filters comes down to requiring that H p (z) is lossless, i.e., stable all-pass. We recall that
for an arbitrary all-pass system G(z) of size p × p it holds that:
G(e iω ) † G(e iω ) = I p , ∀ω ∈ R. (5.26)
Conversely, if (5.26) holds then G(z) is all-pass. If G(z) satisfies the following properties:
G(z) † G(z) = I p for |z| = 1, (5.27)
G(z) † G(z) ≥ I p for |z| < 1, (5.28)
G(z) † G(z) ≤ I p for |z| > 1, (5.29)
where each two equations imply the remaining third one, then G(z) is also stable, thus
lossless.
For any proper rational matrix function R(z) we introduce the following notation:
R † (z) = R(z) † = R † 0 + R† 1 z−1 + R † 2 z−2 + . . . (5.30)
For z = e iω it holds that R † (z −1 ) = R(z) † . Consequently an all-pass system G(z) has
the property that an inverse exists and is given by:
G(z) −1 = G † (z −1 ), (5.31)
for z = e iω , hence for all complex z by analytical continuation (see for example [74]).

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 91
When considering a 2r × 2r lossless polyphase multiwavelet filter H p (z) with multiplicity
r, as described in Section 3.4, it can be partitioned as:
(
)
H 0,e (z) H 0,o (z)
H p (z) =
, (5.32)
H 1,e (z) H 1,o (z)
with H 0,e (z), H 0,o (z), H 1,e (z) and H 1,o (z) the even and odd parts of the low- and highpass
multiwavelet filters, respectively. Due to (3.58) and (5.31) we have that:
(
) (
) ( )
H 0,e (z) H 0,o (z) H † 0,e (z−1 ) H † 1,e (z−1 ) I r 0
H 1,e (z) H 1,o (z) H † 0,o (z−1 ) H † =
. (5.33)
1,o (z−1 ) 0 I r
These conditions must be exactly met in order to generate a valid orthogonal multiwavelet
structure. Although they can be enforced by adding them as constraints to an
optimization routine during the multiwavelet design, this is not convenient as they are
nonlinear and they will often be satisfied merely approximately. A better and more elegant
way is to find a parameterization in which these conditions are built-in. Procedures
for the recursive construction of lossless systems exist in the literature; the procedures
for building all-pass systems from [55] will be used in the remainder of this chapter to
parameterize orthogonal multiwavelets.
A construction method for p × p lossless systems of order n will now be described.
When compact support is required, corresponding to the FIR property of the filters
H 0 (z), H 1 (z) as well as H p (z), special choices can be made to parameterize this subclass
too. Additional properties such as a (balanced) vanishing moment can also be incorporated.
However, the problem of building in vanishing moments of higher order, and in
fact of finding the appropriate balancing conditions, is currently not completely solved
[95, 27, 77, 76, 109, 23, 108].
A key result which underlies the parameterization procedure for lossless systems is the
following. From [55, Proposition 3.2] it follows that if the realization matrix associated
with a realization (A, B, C, D) of some rational function G(z) is unitary, then G(z) will
be lossless. If A is additionally asymptotically stable then (A, B, C, D) will actually be
a minimal and balanced representation. Recall that a balanced realization satisfies the
Lyapunov-Stein equations (4.43)-(4.45), while in the balanced lossless case it holds that
P = Q = I n .
Theorem 5.3.1. Let (A, B, C, D) (with dimensions n × n, n × p, p × n and p × p
respectively) be a balanced state-space realization of an n th order lossless p-input, p-output
transfer function G(z) = D + C (zI n − A) −1 B. Then the (p + n) × (p + n) realization
( ) D C
matrix R =
is unitary.
B A
Conversely, let R be such a unitary block-partitioned matrix, then G(z) = D +
C (zI n − A) −1 B is p × p lossless of degree less than or equal to n. It is of degree n
if and only if A is asymptotically stable, in which case the realization is balanced.
For a proof see [56, 55]. This theorem allows the problem of parameterizing lossless
functions to be studied in terms of unitary realization matrices, associated with balanced

92 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
realizations of lossless functions. Such approach was first carried out in [56] for scalar
(1 × 1) lossless transfer functions, where a balanced canonical form was constructed with
a corresponding upper triangular reachability matrix. The associated realization matrix
allows for a factorization into a product of n more simple unitary building blocks.
This approach was generalized in [55] to multivariable (p × p) lossless functions transfer
functions of degree n. There, it is described how to parameterize such lossless systems
using the tangential Schur algorithm. The associated canonical forms (A, B, C, D) are
parameterized with the associated Schur parameter vectors, while other quantities serve
to index local coordinate charts. In the multivariable case instead of unitary matrix
multiplications, so-called linear fractional transformations are employed to set up a recursive
procedure to construct the realization matrix. It was also found for which choices
in the tangential Schur algorithm this procedure in fact reduces to unitary matrix multiplications
for realization matrices, which is useful from a practical viewpoint. These
procedures will now be described, where the following definitions are helpful.
On the level of transfer functions, for a proper rational matrix G(z), we introduce a
mapping F U,V :
F U,V : G(z) → ˜G(z) (5.34)
defined by:
where
(
F 1 (z)
F 3 (z)
˜G(z) = F 1 (z) + F 2(z)F 3 (z)
z − F 4 (z) , (5.35)
⎛
)
F 2 (z)
= V
F 4 (z) ⎜
⎝
1 0 . . . 0
0
.
0
G(z)
⎞
⎟
⎠ U † , (5.36)
and U and V are (p + 1) × (p + 1) matrices, F 1 (z) is a p × p matrix, F 2 (z) a p × 1 vector,
F 3 (z) a 1 × p vector and F 4 (z) a scalar.
Additionally, it is proven in [55] that if G(z) is proper then ˜G(z) will be well-defined.
Note that lossless transfer functions are always proper.
When using unitary matrices U and V , the mapping F U,V (G(z)) takes lossless functions
to lossless functions. In fact, it constructs a lossless transfer function ˜G(z) with an
order at most one larger than the order of the lossless system G(z).
Theorem 5.3.2. Let U, V be two unitary (p + 1) × (p + 1) matrices and let G(z) be
p × p lossless of degree n. Then ˜G(z) = F U,V (G(z)) is also lossless of degree less than or
equal to n + 1.
For a proof see [55].
On the level of balanced state-space realizations of lossless transfer functions, the
mapping F U,V can be implemented as described by the following result (see again [55]):
Theorem 5.3.3. Let U, V be two unitary (p+1)×(p+1) matrices and let G(z) be p×p
lossless of degree n, with a balanced realization (A, B, C, D). Then ˜G(z) = F U,V (G(z))

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 93
has the realization (Ã, ˜B, ˜C, ˜D) given by:
( ) ⎛ ⎞
1 0 0 (
V 0 ⎜ ⎟
⎝ 0 D C ⎠
0 I n
0 B A
) (
U † 0 ˜D ˜C
=
0 I n
˜B Ã
)
, (5.37)
where Ã, ˜B, ˜C, ˜D respectively have dimensions (n + 1) × (n + 1), (n + 1) × p, p × (n + 1)
and p × p; hence the state-space dimension is n + 1. It is minimal and balanced if and
only if à happens to be asymptotically stable.
This theory allows for a parameterization of lossless systems in state-space by iteratively
applying unitary matrix multiplications. It is also possible to parameterize lossless
systems in terms of transfer functions by using so-called “linear fractional transformations”
as in the more general framework of interpolation theory, more specifically using
the tangential Schur algorithm. A lossless function g(z) increases in McMillan degree by
exactly one into a lossless function ˜G(z), in each step of this tangential Schur algorithm.
The tangential Schur algorithm employs linear fractional transformations (LFT) associated
with J-inner matrices. In [55] LFTs are considered of the form:
(
Θ1 Θ
where Θ =
2
T Θ : G → (Θ 4 G + Θ 3 )(Θ 2 G + Θ 1 ) −1 , (5.38)
Θ 3 Θ 4
)
is a 2p×2p block-partitioned rational matrix function of McMillan
degree m, with blocks of size p × p and G is a p × p rational matrix of McMillan degree
n.
Theorem 5.3.4. If G(z) is p × p lossless of degree n and Θ(z) is 2p × 2p J-inner of
degree m, then ˜G(z) = T Θ(z) (G(z)) is p × p lossless of degree less than or equal to n + m.
Here the function Θ(z) is called J-inner if it holds that:
Θ(z) † JΘ(z) = J for |z| = 1, (5.39)
Θ(z) † JΘ(z) ≤ J for |z| < 1, (5.40)
Θ(z) † JΘ(z) ≥ J for |z| > 1, (5.41)
( )
Ip 0
with J =
. For a proof see [55, Proposition 4.4]. Note that the inverse of a
0 −I p
J-inner function Θ(z) is given by:
Θ(z) −1 = JΘ † (z −1 )J. (5.42)
In the tangential Schur algorithm, a rational p×p lossless function of McMillan degree
n is reduced in n recursion steps to such a function of degree 0; that is, to a constant
unitary matrix. The degree is reduced by m = 1 in each step and the J-inner functions
associated with these steps are also of McMillan degree one. Such function are called
“elementary J-inner factors”.

94 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
The elementary J-inner factors that are employed in [55] are those which have their
pole outside the closed unit disk at z = ¯w −1 . They are represented as:
⎛
⎛(
Θ(u, v, w, ξ, M)(z) =
⎜
⎝ I 2p + ⎝
z−w
1− ¯wz
(
ξ−w
1− ¯wξ
[ ] [ ⎞
) ⎞ u u
J
v v]†
) − 1⎠
(‖u‖ 2 − ‖v‖ 2 ) ⎟ M. (5.43)
⎠
In this expression, the matrix M ∈ 2p×2p is a J-unitary constant matrix, u ∈ C p×1 with
‖u‖ = 1 is a normalized direction vector, v ∈ C p×1 with ‖v‖ < 1 is a Schur vector, w ∈ C
with |w| < 1 is an interpolation point and ξ ∈ C with |ξ| = 1 is a normalization point on
the unit circle. Note that Θ(u, v, w, ξ, M)(ξ) = M.
From [55, Proposition 5.3] we have that:
Theorem 5.3.5. Let G(z) be p × p lossless of McMillan degree n, then
˜G(z) = T Θ(u,v,w,ξ,H)(z) (G(z)) is p × p lossless of McMillan degree n + 1, satisfying the
interpolation condition ˜G( ¯w −1 )u = v.
Conversely from [55, Proposition 5.4] we have:
Theorem 5.3.6. Let ˜G(z) be p × p lossless of McMillan degree n + 1, let w ∈ C be such
that |w| < 1, let u ∈ C p+1 with ‖u‖ = 1 be such that v := ˜G( ¯w −1 )u satisfies ‖v‖ < 1.
Let ξ ∈ C satisfy |ξ| = 1 and M ∈ C 2p×2p be constant J-unitary. Then there exists a
unique lossless p × p function G(z) of McMillan degree n such that ˜G(z) can be obtained
as ˜G(z) = T Θ(u,v,w,ξ,H)(z) (G(z)).
In each step of the parameterization procedure, the order is increased by one, by
choosing values for w, u, ξ and M and by letting v vary over the space of vectors in C p
with ‖v‖ < 1. The interpolation conditions make clear that this gives us coordinate charts
for the space, i.e. the manifold, of p × p lossless functions of degree n. Multiple charts
are needed to cover the manifold; it can be shown that no single Euclidean coordinate
chart is capable of covering the manifold entirely.
In our implementation, a lossless system is preferably built in state-space using a
recursion, i.e., the space of p × p lossless function is parameterized by application of
the reversed tangential Schur algorithm, carried over to the state-space level, starting
from a constant p × p unitary matrix. To achieve this, we describe which choices in the
tangential Schur algorithm allow an LFT ˜G(z) = T Θ(u,v,w,ξ,H)(z) (G(z)) to be represented
as a mapping ˜G(z) = F u,v (G(z)) for suitable unitary matrices U and V . This is discussed
in [55, Theorem 6.4].
Theorem 5.3.7. If T Θ(u,v,w,ξ,M)(z) coincides with a mapping F U,V with U and V unitary,
then
( ) P 0
Θ(u, v, w, ξ, M)(z) = H(uv † )S u,w (z)H(wuv † )
, (5.44)
0 Q

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 95
for some u, v ∈ C p with ‖u‖ = 1, ‖v‖ < 1, some w ∈ C with |w| < 1, and some p × p
unitary matrices P and Q. Equivalently it holds that
( ) P 0
M = H(uv † )S u,w (ξ)H(wuv † )
. (5.45)
0 Q
( )
( )
1 0
In that case one can take U = Û
and V =
0 P
̂V 1 0
, where
0 Q
Û =
̂V =
⎛
⎜
⎝
⎛
⎜
⎝
√
1−|w| 2
√
1−|w| 2 ‖v‖ 2 u I p − (1 + w√ 1−‖v‖ 2
√
1−|w| 2 ‖v‖ 2 )uu†
w √ 1−‖v‖ 2
√
1−|w|2 ‖v‖ 2 √
1−|w| 2
√
1−|w|2 ‖v‖ 2 u†
√
1−|w| 2
√
1−|w|2 ‖v‖
√ v 2 1−‖v‖ 2
√
1−|w| 2 ‖v‖ 2
√
1−‖v‖ 2
I p − (1 − √ ) vv†
1−|w|2 ‖v‖ 2
−
√
1−|w| 2
√
1−|w| 2 ‖v‖ v† 2
‖v‖ 2
⎞
⎟
⎠ , (5.46)
⎞
⎟
⎠ . (5.47)
In this theorem, the notation H(E) is used to denote the Halmos extension of a
strictly contractive p × p matrix E:
( )
(Ip − EE † ) −1/2 E(I
H(E) =
p − E † E) −1/2
E † (I p − EE † ) −1/2 (I p − E † E) −1/2 . (5.48)
It holds that H(E) is Hermitian, J-unitary and invertible with inverse H(E) −1 = H(−E).
Also, the matrix function S u,w (z) is defined by:
( ( ) )
z−w
I p +
S u,w (z) :=
1−wz − 1 uu † 0
. (5.49)
0 I p
For lossless FIR systems, the special case w = 0 is important. In that case:
( )
u Ip − uu †
Û =
0 u † , (5.50)
(
v I p − (1 − √ )
1 − ‖v‖
̂V =
2 )
√ vv†
‖v‖ 2
. (5.51)
1 − ‖v‖
2
−v †
In the following subsection we will discuss the use of the tangential Schur algorithm
to parameterize scalar wavelets. It is shown how to recover the product formula (5.7) by
making suitable choices in the algorithm.
5.3.2 Parameterization of scalar wavelets
In Section 5.2 a lattice filter implementation of 2 × 2 lossless systems was presented. The
associated parameterization procedure employs the product formula (5.7), which can be
generated through a recursion which involves premultiplication in each iteration step:
G (k) (z) = R k Λ(z)G (k−1) (z) (5.52)
G (1) (z) = R 1 Λ(−1). (5.53)

96 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Here, the function G (k) (z) denotes a lossless function of degree k−1, parameterized by the
k parameters θ 1 , θ 2 , . . . , θ k . When attempting to rewrite this in terms of the LFTs and the
tangential Schur algorithm of the previous subsection, it is noted that postmultiplication
is more conveniently represented in that formalism than premultiplication. For that
reason, we address the transposed expressions, which are given by:
(
T (
T
G (z)) (k) = G (z)) (k−1) Λ(z)R
T
k (5.54)
(
T
G (z)) (1) = Λ(−1)R1 T . (5.55)
From these equations it is not difficult to read off a valid interpolation condition and to
construct an associated J-inner matrix Θ (k) (z) which allows one to rewrite an iteration
step in terms of an LFT:
(
( T ( ) ) T
G (z)) (k) = TΘ (k) (z) G (k−1) (z) , (5.56)
for
Θ (k) (z) =
( (Λ(z)R ) )
T −1 ( )
k
0 Rk Λ(z −1 ) 0
=
0 I 2 0 I 2
(5.57)
The following choices can then be made for the parameters in (5.43), with the recursion
index k running from k = 2 to k = n, to recover (5.7) for the scalar wavelet
case:
u k =
( ) sin(θk )
− cos(θ k )
(5.58)
v k =
( 0
0)
(5.59)
w k = 0 (5.60)
ξ k = 1 (5.61)
⎛
⎞
cos(θ k ) − sin(θ k ) 0 0
M k = ⎜sin(θ k ) cos(θ k ) 0 0
⎟
⎝ 0 0 1 0⎠ . (5.62)
0 0 0 1
M k is a block diagonal matrix with the unitary
(
matrices P k = R k and Q k = I 2 on
0
its diagonals. Choosing w k = 0 and v k = in each recursion step, implies that
0)
(
G (k) (z) ) T
has all its poles at z = 0, thus ensuring the FIR property for this polyphase
filter. (This corresponds to a Potapov decomposition of ( G (k) (z) ) T
.) The parameter θk
shows up in the normalized direction vector u k , which gives a non-standard choice for
parameterizing lossless functions. The standard choice to parameterize lossless functions
requires u k , w k , ξ k and M k to be fixed and v k to contain the free parameters subject to

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 97
the constraint ‖v k ‖ < 1. The current non-standard choice, where v k is kept fixed and
u k is varied, does produce a parameterization however which works properly, although
some lossless functions can be obtained for different choices of parameters.
It is now straightforward to verify that the LFT in each recursion step admits a
representation as a mapping F U,V . Referring to Theorem 5.3.7 we have that:
⎛ √ ( √ ) ⎞
1−|wk |
√ 2
⎜
u 1−|wk | 2 ‖v k ‖ 2 k I p − 1 + w k 1−‖vk ‖
√ 2
u k u †
1−|wk | 2 ‖v k ‖ 2 k⎟
Û =
=
̂V =
=
⎝
√ √
w k 1−‖vk ‖
√ 2
1−|wk |
√ 2
1−|wk | 2 ‖v k ‖ 2 1−|wk | 2 ‖v k ‖ u† 2 k
⎛
sin(θ k ) cos 2 ⎞
(θ k ) sin(θ k ) cos(θ k )
⎝− cos(θ k ) sin(θ k ) cos(θ k ) sin 2 (θ k ) ⎠ . (5.63)
0 sin(θ k ) − cos(θ k )
⎛ √ ( √ ) ⎞
1−|wk |
√ 2
⎜
1−‖vk ‖
1−|wk |
⎝
‖v k ‖ 2 k I p − 1 + √ 2 v k v † k
1−|wk | 2 ‖v k ‖ 2 ‖v k ‖
√ 2
⎟
⎠
1−‖vk ‖
√ 2
− 1−|w k |
√ 2
1−|wk | 2 ‖v k ‖ 2 1−|wk | 2 ‖v k ‖ v† 2 k
⎛ ⎞
0 1 0
⎝0 0 1⎠ . (5.64)
1 0 0
The matrices U and V that are used in the mappings F U,V in the steps of the
corresponding state-space recursion (5.37) are given by:
⎛
⎞ ⎛
⎞
1 0 0
sin(θ k ) cos(θ k ) 0
U = Û ⎜
⎟ ⎜
⎟
⎝ 0
⎠ = ⎝ − cos(θ k ) sin(θ k ) 0 ⎠ (5.65)
P k
0
0 0 −1
⎛
⎞ ⎛ ⎞
1 0 0
0 1 0
V = ̂V ⎜
⎟ ⎜ ⎟
⎝ 0
⎠ = ⎝ 0 0 1 ⎠ . (5.66)
Q k
0
1 0 0
The resulting state-space recursion (5.37) then attains the following form:
(
)
B (k) A (k)
D (k) C (k)
⎛
cos(θ k )D (k−1)
1,1 sin(θ k )D (k−1)
1,1 −D (k−1)
1,2
C (k−1)
1,1 . . . C (k−1)
1,k−1
cos(θ k )D (k−1)
2,1 sin(θ k )D (k−1)
2,1 −D (k−1)
2,2
=
⎜
⎝
.
cos(θ k )B (k−1)
k−1,1
⎠
C (k−1)
2,1 . . . C (k−1)
2,k−1
sin(θ k ) − cos(θ k ) 0 0 . . . 0
cos(θ k )B (k−1)
1,1 sin(θ k )B (k−1)
1,1
.
sin(θ k )B (k−1)
k−1,1
−B (k−1)
1,2
.
−B (k−1)
k−1,2
A (k−1)
⎞
⎟
⎠
(5.67)

98 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
From the previous equation it can be seen that the dynamical matrix A (k−1) is extended
by means of a zero row and a first column that is equal to the second column of
B (k−1) . Hence A (k) is strictly lower triangular, having a zero diagonal. Consequently its
eigenvalues are zero and the system is FIR.
Note that (5.67) yields a balanced realization of G (k) (z) T and it may be transposed
in order to obtain a balanced realization of G (k) (z). It follows that A (k)T is a strictly
upper triangular matrix: it is in real Schur form. (Balancing leaves the freedom of the
orthogonal group which allowed us to bring A (k)T into that real Schur form.) Since the
real Schur form is not necessarily unique it is not a canonical form; this is a consequence
of the non-standard parameterization mentioned earlier.
5.3.3 Parameterization of multiwavelets
The parameterization in the previous section can be extended to the multiwavelet case.
For the parameters of the elementary J-inner factors from (5.43) the following choices
are made: Since we are building FIR filters we take w = 0 and v = (0 . . . 0) T . The
parameter ξ can be chosen freely on the unit circle; the choice ξ = 1 is made so that
S u,w (ξ) = I 4r . In order to again ensure that T Θ(u,v,w,ξ,M)(z) coincides with a mapping
F U,V as in Theorem 5.3.7, due to the convenient choice of ξ we need to have that M =
( ) P 0
.
0 Q
Theorem 5.3.8. Let G (k+1) (z) be a FIR all-pass filter of order k + 1. Then with the
choice of w k = 0, v k = (0 . . . 0) T , ‖u k ‖ = 1, ξ k = 1, P and Q unitary, it follows that
G (k+1) (z) can be factored as:
G (k+1) (z) = (I 2r + (z −1 − 1)u k u † k )P kG (k) (z)Q † k . (5.68)
where G (k) (z) is a FIR all-pass filter of order k.
Proof. Obtaining a state-space recursion for a block diagonal matrix G k involves a factorization
of Θ (k) (z) as in [55, Proposition 5.2]:
Θ (k) (u k , v k , w k , ξ k , M k )(z) = H(u k v † k )S u k ,w k
(z)S uk ,w k
(ξ k ) −1 H(u k v † k )−1 M k . (5.69)
Also observe that for the choices made [55, Theorem 6.4] holds, such that:
( )
Θ (k) (u k , v k , w k , ξ k , M k )(z) = ̂Θ (k) Pk 0
(u k , v k , w k )(z)
, (5.70)
0 Q k
with
̂Θ (k) (u k , v k , w k )(z) = S uk ,0(z) =
(
)
I 2r + (z − 1)u k u † k
0
. (5.71)
0 I 2r
Using the LFT formula in (5.38) we see that the inverse of the upper right quadrant of
(5.71) is used as a post multiplying factor. However we are working on the transposed

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 99
system as noted in (5.56), such that after transposition the inverse factor appears as a
pre-multiplying factor. This leads to the recursion formula:
( )
G (k+1) (z) T = T Θ(uk ,v k ,w k ,ξ k ,M k )(z) G (k) (z) T = Q k G (k) (z) T P † k (I 2r + (z −1 − 1)u k u † k ),
(5.72)
such that:
G (k+1) (z) = (I 2r + (z −1 − 1)u k u † k )P kG (k) (z)Q † k
(5.73)
Observe that this is a more general parameterization than previously available in the
literature. One can obtain a lattice decomposition in terms of Givens rotation matrices
from Theorem 5.3.8 with the following choices:
We relate the unitary matrix P to the vector u, which will again contain the parameters.
Analogous to the scalar case we require that P † u = −e 2r and any such P will do.
Here e n denotes the n th standard basis vector. As in the scalar case we choose Q = I 2r .
For the parameter vector u it must again hold that ‖u‖ = 1 and as a choice for a
fully parameterized u one can for example make:
⎛
⎞
sin(α 1 ) sin(α 2 ) sin(α 3 )
u = ⎜− cos(α 3 ) sin(α 1 ) sin(α 2 )
⎟
⎝ cos(α 2 ) sin(α 1 ) ⎠ . (5.74)
− cos(α 1 )
Corollary 5.3.9. Let G (k+1) (z) be a FIR all-pass filter of order k + 1. Then with the
choice of w k = 0, v k = (0 . . . 0) T , ‖u k ‖ = 1, ξ k = 1, Q = I 2r and P † k u k = −e 2r it follows
that G (k+1) (z) can be factored in a lattice decomposition as:
⎛
⎞
1
.
G (k+1) .. (z) = P k ⎜
⎟
⎝ 1 ⎠ G(k) (z), (5.75)
z −1
where G (k) (z) is a FIR all-pass filter of order k.
Proof. Observe that due to the fact that P k is a unitary 2r × 2r matrix and that P k u k =
−e 2r , the following equalities hold:
Such that (5.68) can be rewritten as:
u k = −P k e 2r , (5.76)
u † k P k = −e † 2r P † k P k = −e 2r , (5.77)
G (k+1) (z) = P k (I 2r + (z −1 − 1)e 2r e † 2r )G(k) (z)Q † k
(5.78)
⎛
1
. .. = P k ⎜
⎟
⎝ 1 ⎠ G(k) (z)Q † k . (5.79)
z −1 ⎞

100 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Since Q k = I 2r the following product formula is obtained:
⎛
⎞
1
.
G (k+1) .. (z) = P k ⎜
⎟
⎝ 1 ⎠ G(k) (z), (5.80)
z −1
which completes the proof.
A practical choice for P relating to the choice of u as in (5.74) is to build it as a
product of Givens rotation matrices as in for example [116]. Such a parameterization for
r = 2 is given by:
⎛
⎞ ⎛
⎞
cos(α 3 ) − sin(α 3 ) 0 0 1 0 0 0
P (α 1 , α 2 , α 3 ) = ⎜sin(α 3 ) cos(α 3 ) 0 0
⎟ ⎜0 cos(α 2 ) − sin(α 2 ) 0
⎟
⎝ 0 0 1 0⎠
⎝0 sin(α 2 ) cos(α 2 ) 0⎠
0 0 0 1 0 0 0 1
⎛
⎞
1 0 0 0
⎜0 1 0 0
⎟
⎝0 0 cos(α 1 ) − sin(α 1 ) ⎠
0 0 sin(α 1 ) cos(α 1 )
This results in the following parameterized matrix P :
P =
( cos(α3) − cos(α 2) sin(α 3) cos(α 1) sin(α 2) sin(α 3) − sin(α 1) sin(α 2) sin(α 3)
sin(α 3) cos(α 2) cos(α 3) − cos(α 1) cos(α 3) sin(α 2) cos(α 3) sin(α 1) sin(α 2)
0 sin(α 2) cos(α 1) cos(α 2) − cos(α 2) sin(α 1)
0 0 sin(α 1) cos(α 1)
)
, (5.81)
which indeed satisfies the condition P † k u k = −e 2r .
The parameterization as in the so-called “great factorization theorem”, cf. [107, 117],
can be obtained with the choice of P k = Q k = I 2r .
Corollary 5.3.10. Let G (k+1) (z) be a FIR all-pass filter of order k + 1. Then with the
choice of w k = 0, v k = (0 . . . 0) T , ‖u k ‖ = 1, ξ k = 1, Q = I 2r and P = I 2r it follows that
G (k+1) (z) can be factored in a Householder decomposition as:
(
G (k+1) (z) = I 2r − u k u k + u k u † k z−1) G (k) (z), (5.82)
where G (k) (z) is a FIR all-pass filter of order k.
Proof. Consider (5.68) from Theorem 5.3.8. With the choice of P k = Q k = I 2r , (5.68)
can be rewritten as:
(
G (k+1) (z) = I 2r − u k u k + u k u † k z−1) G (k) (z). (5.83)

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 101
Using the parameterization in Corollary 5.3.9 of multiwavelets as polyphase all-pass
systems all requirements but the first (balanced) vanishing moment are enforced. In the
next section it is discussed how a first balanced vanishing moment can be built in as
first discussed in [95]. It will turn out that the first balanced vanishing moment can be
enforced by initializing the recursion in a specific way. In Section 5.3.5 a step-by-step
design procedure for orthogonal multiwavelets will be provided.
5.3.4 Balanced vanishing moments
When considering a constant signal, we may investigate how it is processed when employing
Mallat’s cascade algorithm [79]. For scalar wavelets, the approximation coefficients
will all be identical, i.e., a constant coefficient vector. Therefore the sampled signal is
commonly used to initialize the multi resolution analysis, because the sampled signal
and the approximation coefficients constitute similar sequences. This avoids an actual
computation of the scaling coefficients that express the input signal in terms of the basis
spanned by shifted versions of the scaling function to initialize the multi resolution
analysis and it also avoids the actual computation of the multiscaling and multiwavelet
function.
When processing a constant signal in the multiwavelet case, one finds that each subsequence
of scaling coefficients, for each of the multiwavelets, is also constant. However,
when these constants differ per multiwavelet, then one cannot simply use the sampled signal
to initialize the multi resolution analysis. The necessary condition to use a sampled
signal to initialize the multiresolution analysis is that the low-pass synthesis operator
should exactly preserve polynomials of order p for a p th order balanced vanishing moment
[77]. If this condition were not satisfied a preprocessing step would be required,
which is unattractive. The alternative is to impose a so-called balancing condition on
the multiwavelet structure and to allow again the direct use of the sampled signal.
A 0 th order balanced scaling function preserves constant signals [27]:
∫
R
∫
1φ [0]
(t)dt = . . . =
R
1φ [r−1]
(t)dt. (5.84)
Due to power complementarity this is equivalent to the condition that the detail coefficients
are not affected by constant signals: 0 = ∫ R 1ψ (t)dt. When taking the dilation
[k]
equation (3.24) and wavelet equation (3.25) and integrating both sides over the real axis
we obtain:
ζ 0 =
η 0 =
∫
∫
R
R
φ(t)dt = √ 2n−1
∑
∫
2 C k
k=0
ψ(t)dt = √ 2n−1
∑
∫
2 D k
k=0
R
R
φ(2t + k)dt, (5.85)
φ(2t + k)dt. (5.86)
The desired vanishing moment comes down to the condition η 0 = 0. The balancing

102 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
⎛ ⎞
1
⎜
condition comes down to ζ 0 = c
. ⎟
⎝.
⎠, for some constant scalar c. Using the change of
1
basis τ = 2t + k, dτ = 2dt and dt = 1 2dτ we obtain:
∫
∫
φ(2t + k)dt = φ(τ) 1 2 dτ = 1 2 ζ 0, (5.87)
from which it follows that:
R
R
ζ 0 = 1 2√
2H0 (1)ζ 0 , (5.88)
η 0 = 1 2√
2H1 (1)ζ 0 . (5.89)
This can be rewritten in terms of the polyphase matrix H p (z) as:
( )
ζ0
= 1 ( )
√ ζ0
2Hp (1) . (5.90)
η 0 2 ζ 0
Since H p (z) is all-pass and thus orthogonal at z = 1 we have that:
( )
1
2 ‖H ζ0
p(1) ‖ 2 = 1 ( )
ζ 0 2 ‖ ζ0
‖ 2 = ‖ζ 0 ‖ 2 , (5.91)
ζ 0
( )
ζ0
‖ ‖ 2 = ‖ζ 0 ‖ 2 + ‖η 0 ‖ 2 , (5.92)
η 0
and since these expressions are equal due to (5.90) it follows that‖η 0 ‖ 2 = 0 and η 0 = 0.
It can then be concluded that the first vanishing moment is built-in, meaning that every
feasible (multi)wavelet structure which fits the orthogonal set-up as discussed previously
in this chapter, already obeys the conditions for a first vanishing moment. The conclusion
here is that the first vanishing moment is implicitly required and restricts the class of
lossless polyphase matrices H 0 (z), H 1 (z). The conditions for a first balanced vanishing
moment can be enforced in the tangential Schur algorithm by means of an interpolation
condition on the unit circle. The condition on H p
(k) (z) for the first balanced vanishing
moment is as in [95]:
Theorem 5.3.11. If H p
(k) (z) is a 2r×2r real FIR all-pass filter of order k associated with
the corresponding FIR filters H 0 (z), H 1 (z) and vector functions φ(t) and ψ(t), satisfying
the dilation an wavelet equation, and √ 2 is a simple eigenvalue of H 0 (1), then ψ(t) has
a balanced vanishing moment of order 0 if and only if
(1, 1, . . . , 1|1, 1, . . . , 1) H (k)
p (1) T = √ 2 (1, 1, . . . , 1|0, 0, . . . , 0) (5.93)
Proof. This theorem directly follows from (5.90). Note that the condition that √ 2 is
a simple eigenvalue of H 0 (1) ensures that (5.88) has a solution ζ 0 which is determined
up to a nonzero scaling factor. Orthogonality of the multiresolution structure and sign
conventions for φ(t) and ψ(t) fix this scaling factor.

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 103
The vanishing moment condition is now rewritten in terms of H 0 (z), H 1 (z). The
following relation holds:
( ) ( )
1√ 2Hp (z 2 Ir 0 Ir I
) r
2
0 z −1 = 1 ( )
√ H0 (z) H 2 0 (−z)
. (5.94)
I r I r −I r 2 H 1 (z) H 1 (−z)
This equation allows one to relate the derivatives of H p (z) to the derivatives of H 0 (z),
H 1 (z), H 0 (−z) and H 1 (−z), in particular at z = 1. Additionally, due to orthogonality,
we have that H 1 (1) = 0, H 1(1) ′ = 0, H 1 ”(1) = 0 is equivalent to H 0 (−1) = 0, H 0(−1) ′ =
0, H 0 ”(−1) = 0. This specifies the vanishing moment condition entirely in terms of the
low-pass filter. Note that all matrices on the left hand side of (5.94) are real lossless
matrices, hence the matrix on the right hand side is real lossless and thus orthogonal at
|z| = 1 and more specifically at z = 1:
(
1√ H0 (1) 2
2 H 1 (1)
)
H 0 (−1)
H 1 (−1)
(
1√ Ir
= H p (1)I 2r 2
2
)
I r
I r −I r
. (5.95)
Using the right hand side of (5.95) and that η 0 = 0, (5.90) can be rewritten as:
( )
ζ0
= H p (1) 1 ( ) ( )
√ Ir I 2 r ζ0
0 2 I r −I r 0
= 1 2
√
2
(
H0 (1)
H 0 (−1)
H 1 (1) H 1 (−1)) (
ζ0
0
(5.96)
)
. (5.97)
One can investigate how this condition translates to H p (1) in the scalar case, i.e.,
√
r = 1. The conditions become: 2ζ0 = H 0 (1)ζ 0 and 0 = H 1 (1)ζ 0 . Since ζ 0 ≠ 0, it
must hold that H 0 (1) = √ 2 and H 1 (1) = 0 and due to power complementarity that
H 0 (−1) = 0 and thus that H 1 (−1) = ± √ 2. The sign choice for H 1 (−1) corresponds to
a sign choice for the wavelet ψ(t). We shall use the convention H 1 (−1) = √ 2. Hence
( )
√ 1 1
H p (1) = 1 2 2 . In the multiwavelet case balancing is required and we have to
1 −1
work with the scalar multiples.
Using (5.97), the condition for the first (0 th order) vanishing moment becomes [77]:
√
T
2 (1, 1, . . . , 1) = H 0 (1) (1, 1, . . . , 1) T (5.98)
(0, 0, . . . , 0) T = H 1 (1) (1, 1, . . . , 1) T (5.99)
As a side remark it can be verified that the results in this section are consistent with
the condition for a first balanced vanishing moment in [77, Theorem 2]. Taking into
account the differences in notation and terminology the balanced vanishing moment is
characterized in [77] by the conditions:
(1, 1, . . . , 1) H 0 (1) = √ 2 (1, 1, . . . , 1) (5.100)
(1, 1, . . . , 1) H 0 (−1) = √ 2 (0, 0, . . . , 0) (5.101)
The conditions in (5.100) and (5.101) relate to the conditions in (5.98) and (5.99) due to
the following lemma, which is a application to (5.93):

104 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
Lemma 5.3.12. Consider an orthogonal real matrix N with a real eigenvalue λ and a
corresponding real eigenvector v. Then the equality Nv = λv is equivalent to v T N = λv T .
Proof. Consider the following equality:
The norm on both sides needs to be equal as well:
Nv = λv (5.102)
v T N T Nv = |λ| 2 v T v, (5.103)
v T v = |λ| 2 v T v, (5.104)
hence |λ| 2 = 1 and thus for real eigenvalues we have λ = 1 or λ = −1. Now construct an
orthogonal matrix N 2 which has v as its first column: v = N 2 e 1 . We can then rewrite
(5.102) as:
NN 2 e 1 = λN 2 e 1 (5.105)
N T 2 NN 2 e 1 = λe 1 (5.106)
Due to the fact that |λ| = 1, the orthogonal matrix N2 T NN 2 will have the following
structure:
⎛
⎞
λ 0 . . . 0
0
N2 T NN 2 =
⎜
⎟
⎝ . N3 ⎠ ,
0
where N 3 is another orthogonal matrix. Hence we can write:
which completes the proof.
e T 1 N T 2 NN 2 = λe T 1 , (5.107)
e T 1 N T 2 N = λe T 1 N T 2 , (5.108)
v T N = λv T , (5.109)
A consequence of this lemma is that the transpose in (5.93) can be omitted.
The fact that the parameterization in Corollary 5.3.9 has the convenient property
that H p
(k+1) (1) = H p
(k) (1), makes it possible to build in a vanishing moment with respect
to the multiwavelet in a straightforward manner.
Theorem 5.3.13. An orthogonal multiwavelet of multiplicity r having a balanced vanishing
moment and a FIR filter of order k can be built as a polyphase all-pass H p
(k) (z)
by initializing the factorization in Corollary 5.3.9 with a zeroth order polyphase all-pass
H p
(0) that is factored as:
( ) 1 0
H p (0)T = H α H β , (5.110)
0 L

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 105
where H α and H β are fixed orthogonal Householder matrices of the form:
with:
H α = I 2r − 2 ααT
α T α,
(5.111)
α T = (1, 1, . . . , 1|0, 0, . . . , 0) / √ r − (1, 0, . . . , 0|0, 0, . . . , 0) (5.112)
β T = (1, 1, . . . , 1|1, 1, . . . , 1) / √ 2r − (1, 0, . . . , 0|0, 0, . . . , 0) , (5.113)
and L is an arbitrary (2r − 1) × (2r − 1) orthogonal matrix.
Proof. First observe that α and β are of the form α =
a−b
‖a−b‖. The Householder reflection
H α x reflects the component of x in the direction a to b. Components that are orthogonal
to a are left unchanged.
Since (5.71) shows that ̂Θ (k) (u k , v k , w k )(1) = I 2r , ∀k interpolation conditions on
H p
(k) (1) that are built-in in H 0 are preserved throughout the recursion. It is now left to
correctly initialize this recursion.
The interpolation condition of interest is the balancing condition from Theorem 5.3.11:
H (0)
p (1, 1, . . . , 1|1, 1, . . . , 1) T = (1, 1, . . . , 1|0, 0, . . . , 0) T . (5.114)
Now suppose that we factor H p (0) as the product of three matrices H p
(0) = W RU, with
W and U symmetrical. Then we get the condition:
W RU (1, 1, . . . , 1|1, 1, . . . , 1) T = (1, 1, . . . , 1|0, 0, . . . , 0) T , (5.115)
which can be rewritten as (using the symmetry property):
RU (1, 1, . . . , 1|1, 1, . . . , 1) T = W (1, 1, . . . , 1|0, 0, . . . , 0) T . (5.116)
If the matrix W is chosen to be H α then the right hand side of (5.116) simplifies to e 1
and similarly if U is chosen to be H β then the left hand side of (5.116) simplifies to Re 1
as a direct consequence of the theory on Householder transformations. Thus with this
choice of W and U any matrix of the following form satisfies the condition in (5.116):
( ) 1 0
. (5.117)
0 L
Since we want to obtain an orthogonal system the initial matrix H p
(0) and thus L has
to be orthogonal. Since it is multiplied with orthogonal matrices during the recursion
orthogonality will be preserved.
Note that such an orthogonal matrix L can be parameterized by (2 − 1)(2r − 1) free
angular parameters in a straightforward way. For example for r = 2 a 3 × 3 matrix Q
can be parameterized as:
( ) ( 1 0 I2 0
=
0 L 0 R(θ 3 )
) ⎛ ⎞
0 0
⎝1
0 R(θ 2 ) 0⎠
0 0 1
( )
I2 0
. (5.118)
0 R(θ 1 )

106 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
( )
cos(θ) − sin(θ)
R(θ) =
sin(θ) cos(θ)
(5.119)
5.3.5 Multiwavelet design
An explicit algorithm for constructing a multiwavelet filter of order n with r = 2 is given
by:
1. Initialize the recursion:
a) Select initial angles θ1, 0 θ2, 0 θ3.
0
( ) 1 0
b) Construct the matrix from (5.118) using parameters θ 0
0 L
1, θ2, 0 θ3.
0
c) Construct the matrices U and W from (5.116) as:
U =
W =
⎛
⎜
⎝
1 1
2 2
1 1
2
1
2
− 1 2
1
d) Construct the zeroth order all-pass H (0)
p
2. Set k = 1 and enter the recursion
3. Recursion step
a) Select angles θ k 1, θ k 2, θ k 3.
1
2
1
2
2
− 1 2
− 1 2⎟
1 ⎠ (5.120)
2
− 1 2
1
2
⎞
2
− 1 2
− 1 2
⎛ √
1
2 2
1 ⎞
2√
2 0 0
√ 1
⎜ 2 2 −
1
2√
2 0 0
⎟
⎝ 0 0 1 0⎠ (5.121)
0 0 0 1
as:
H (0)
p = W RU. (5.122)
b) Use these angles to construct the matrix P k as in (5.81).
c) Recursively construct a k th order multiwavelet filter H p
(k) (z) from a (k − 1) th
order filter in accordance to Corollary 5.3.9:
( )
H p (k) I3 0
= P k
0 z −1 H p (k−1)
(5.123)
4. If k < n then set k = k + 1 and goto 3.
The parameterization of multiwavelets as stable all-pass systems in the previous section
allows for the design of multiwavelets for a specific purpose. The discussion on a
suitable criterion for wavelet design as in Section 5.1 applies here as well since energy
preservation holds when considering the overall multiwavelet filter.

5.3. MULTIWAVELET PARAMETERIZATION AND DESIGN 107
Wavelet coefficients 1
A_6
D_6
200
D_5
D_4
0
D_3
D_2
D_1 -200
83 120 160 230
Wavelet coefficients 2
A_6
400
D_6
D_5
200
D_4
0
D_3
D_2
-200
D_1 -400
83 120 160 230
Prototype signal
400
200
0
83 120 160 230
Figure 5.11: Windowing for multiwavelet design
The orthogonality between the different filters in multiwavelets make them conceptually
an appealing tool to distinguish and detect features simultaneously. In that respect
it is opportunistic to employ r time-scale masks (as in Figure 5.11) to the wavelet coefficients.
So for a multiwavelet with r = 2 two time-scale masks are used such that two
features can be discriminated. When using these masks for optimization the conservation
of energy assumption no longer holds. Also it is not convenient to use a criterion (such
as l 1 minimization) that involves minimization since it is possible that the optimization
routine converges to an optimum that is an effective delay such that features are pushed
outside the mask. It is still possible to use the l 4 maximization, however since we want
to put the energy at a specific place it is also possible to use l 2 maximization. Another
point of interest is that for detection purposes it is most convenient to have full resolution
at each scale, i.e. to use an undecimated multiwavelet transform.
As was the case for scalar wavelet design the optimization is performed by means of
a local search technique with random starting points. The choice of parameterization
ensures that all required conditions in order to obtain an orthogonal stable all-pass with
at least one vanishing moment is built-in. However for applications such as filtering

108 CHAPTER 5. ORTHOGONAL WAVELET DESIGN
additional smoothness might be needed. For the smoothness of multiwavelet functions a
vanishing moment is not sufficient. Additionally the multiwavelet needs to be balanced
[76, 77, 27]. Balancing requires an additional interpolation condition that is difficult
to handle in the current parameterization and is still an open problem. It is still not
guaranteed that the resulting system is indeed a multiwavelet since it is not enforced
that each set of wavelet coefficients forms a high-pass filter and the associated scaling
coefficients form a low-pass filter.

Chapter 6
Biomedical applications of orthogonal
(multi)wavelet design
In this chapter the potential of the approaches, as introduced in Chapter 5, will be
demonstrated by means of a number of practical applications. In Section 6.1 two practical
applications of (multi)wavelet design with respect to cardiac signal processing are
discussed. In Section 6.1.1 it is demonstrated how scalar wavelet design is useful for the
detection of QRS complexes in ECG signals. In Section 6.1.2 the length of the QT interval
is estimated using designed multiwavelets. In Section 6.2 the use of wavelet design for
an application in 2D image analysis is discussed. This describes the removal an artifact
in magnetic resonance imaging, called the “bias field”, using specially designed wavelets
[64].
6.1 Applications of orthogonal (multi)wavelet design in cardiology
6.1.1 Detecting the QRS complex using orthogonal wavelet design
Using the wavelet design approach in Chapter 5 it is possible to design a wavelet for
a given signal of interest. Given an ECG signal (see Section 2.3) one can average the
beats in the signal and use them as a prototype for the wavelet design algorithm. If
an ECG signal consisting of multiple beats was used as a prototype, the wavelet design
would have searched for a wavelet that performs well for a superposition of the beats
in the ECG signal. Hence it is justified to construct a prototype by aligning the ECG
beats. The annotated R peak is used as a reference for the alignment. Since the QRS
complex is the dominant part of an ECG beat, the location of this complex is generally
not hard to detect. However as discussed in Section 2.2, the heart is a 3D organ. As
such, the signals that are measured extracardiacally may exhibit different morphologies,
dependent on the position of the lead, relative to the heart, complicating the detection.
109

110 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
Daubechies 2
1
0.5
0
−0.5
0 1 2 3
L , 4 parameters
1
1
0.5
0
−0.5
0 2 4 6
L , 3 parameters
4
1
0.5
0
−0.5
1
0.5
0
L 1
, 2 parameters
0 1 2 3
L 1
, 5 parameters
−0.5
0 2 4 6 8
L 4
, 4 parameters
1
0.5
0
L 1
, 3 parameters
−0.5
0 2 4
4
2
0
−2
L 4
, 2 parameters
0 1 2 3
L 4
, 5 parameters
1
0.5
0
−0.5
0 2 4
1
0
−1
0 2 4 6
2
1
0
−1
0 2 4 6 8
Figure 6.1: Wavelets used for the detection of the QRS complex in episode 103.
Furthermore patient-to-patient variation and pathologies may affect the morphology of
ECG signals.
In order to assess whether the wavelet design procedure from Section 5.2 indeed yields
a beneficial distribution of the energy in the wavelet domain for the detection of the QRS
complex, the ECG beat from Figure 5.1(a) was used as a prototype signal for wavelet
design for two to five parameters and for both criteria. This prototype is a superposition
of the normal beats from episode 103 of the MIT-BIH arrythmia database [43]. Next
the stationary wavelet transform (see Section 3.3, [91]) was calculated for 2 19 samples of
episode 103 of the MIT-BIH arrythmia database. The choice of this length is a result
of the used implementation of the SWT that accepts only signal with a length that is a
power of two. For the detection step only a single scale was taken into account. Note that
a sparse representation in the wavelet domain may yield a certain pattern in space-scale
that can be used as a detector for a given morphology. A QRS complex was detected
if the absolute value of the wavelet coefficients at that particular scale exceeded a fixed
threshold value. This threshold value was chosen for each wavelet and each scale such that

6.1. APPLICATIONS OF WAVELET DESIGN IN CARDIOLOGY 111
c 0 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9
db2 0.4830 0.8365 0.2241 -0.1294
l 1 , 2p 0.5010 0.8313 0.2061 -0.1242
l 1 , 3p 0.1860 0.7394 0.6336 -0.0607 -0.1125 0.0283
l 1 , 4p -0.1077 0.1915 0.7849 0.5750 0.0167 -0.0668 0.0132 0.0074
l 1 , 5p 0.0159 0.0279 -0.0615 0.1402 0.7140 0.6652 0.0593 -0.1380 -0.0206 0.0117
l 4 , 2p 0.1039 0.7868 0.6032 -0.0797
l 4 , 3p -0.1111 0.1612 0.8158 0.5443 0.0024 0.0017
l 4 , 4p -0.1143 0.2268 0.8451 0.4699 -0.0190 0.0128 -0.0047 -0.0024
l 4 , 5p -0.1017 0.2915 0.8666 0.3894 -0.0311 0.0023 -0.0170 0.0273 -0.0097 -0.0034
Table 6.1: Low-pass filters used for the detection of the QRS complex in episode 103.
The number before “p” indicates the number of parameters used to design the filter.
the sum of the false negatives and false positives was minimized. Since the effectiveness of
the approach is evaluated here and not the robustness of a detector, this is a valid choice
for the goal at hand. If the threshold value was exceeded within 20 samples (55ms) of
an annotated QRS complex this was accepted as a valid detection. If it was outside this
interval a false positive was reported. This is a very basic detection scheme since we do
not aim to find a detector for the QRS complex which is nearly flawless, or even tries to
find the exact location of the complex, but we are assessing the potential of the wavelet
design approach. Note that the false positives may be clotted together; in this case all of
them are still considered individual false positives. As reference wavelet the Daubechies
2 wavelet was chosen due to the fact that it is quite popular (see e.g. [105, 114, 93, 1])
and that the steep slopes in the signal are captured well by the Daubechies 2 wavelet
function.
All designed wavelets and the Daubechies 2 wavelet were able to correctly find all
or all but one of the QRS complexes at level 3 and 4 of the wavelet transform in the
1690 beats of episode 103 that were considered. Upon examining the wavelet and scaling
functions in Figure 6.1, we see that indeed the (possibly mirrored) Daubechies 2 wavelet
was found as an optimum for some of the designed wavelets, even though in some of these
cases a longer filter size was used. From the corresponding low-pass filter coefficients in
Table 6.1, we see that for a large number of higher order filters a number of coefficients
are close to zero and that the remaining coefficients are in the same order of magnitude
as for the Daubechies 2 wavelet. This corresponds to Observation 5.2.7.
Next, as a second test, a dataset was used which exhibits more variation in terms
of morphology. The selected dataset: episode 215 of the MIT-BIH arrythmia database
[43] does not only contain normal beats, but also atrial premature contractions and
premature ventricular contractions. The normal beats as illustrated in Figure 6.2 have a
different morphology than the normal beats of episode 103 as displayed in Figure 5.1(a).
Of episode 215 an excerpt consisting of 2718 beats was considered. The results are
displayed in Table 6.2.
From Table 6.2 it can be seen that the l 1 designed wavelets perform better than the
Daubechies 2 wavelet for this dataset. The results for the l 4 designed wavelets are not as
good. The Daubechies 3 wavelet performs quite well. It can however be seen from the
table that an increase in order for the Daubechies wavelets does not automatically yield
an improvement in terms of detection performance. Among the higher order wavelets,

112 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
200
150
100
50
0
−50
−100
50 100 150 200 250
Figure 6.2: Smoothed normal beat from episode 215 of the MIT-BIH database.
wavelet false neg. false pos. total errors level thres.
Daubechies 2 9 3 12 4 295
Daubechies 3 4 3 7 4 243
Daubechies 4 20 0 20 4 311
Daubechies 5 8 4 12 4 256
Daubechies 6 23 5 28 4 304
Daubechies 7 78 27 105 4 367
Daubechies 8 71 17 88 4 347
l 1 , 2 parameters 8 1 9 4 255
l 1 , 4 parameters 8 3 11 4 271
l 1 , 5 parameters 6 0 6 4 252
l 1 , 6 parameters 4 0 4 4 240
l 1 , 8 parameters 3 0 3 4 230
l 4 , 2 parameters 18 0 18 4 380
l 4 , 3 parameters 9 4 13 4 303
l 4 , 4 parameters 8 4 12 4 294
Table 6.2: Detection of the QRS complex in 2718 beats of episode 215 of the MIT-BIH
arrythmia database. The used wavelet filter, the number of missed beats, the number of
false detections, the total number of errors, the used wavelet scale and the used threshold
value are displayed in the columns respectively.

6.1. APPLICATIONS OF WAVELET DESIGN IN CARDIOLOGY 113
1
0.5
0
Daubechies 2
−0.5
0 1 2 3
L 1
, 5 parameters
1
0.5
0
L 1
, 2 parameters
−0.5
0 1 2 3
L 1
, 6 parameters
1
0.5
0
L 1
, 4 parameters
−0.5
0 5
L 1
, 8 parameters
1
0.5
0
−0.5
0 5
L 4
, 2 parameters
1
0.5
0
−0.5
0 5 10
L 4
, 3 parameters
1
0.5
0
−0.5
0 5 10 15
L 4
, 4 parameters
1
0.5
0
−0.5
0 1 2 3
1
0.5
0
−0.5
0 5
1
0.5
0
−0.5
0 5
Figure 6.3: Wavelets used for the detection of the QRS complex in episode 215 of the
MIT-BIH database.
the designed wavelets tend to perform better in this table. Summarizing, for low-order
wavelet filters, the property to have vanishing moments may thus be beneficial in a
morphologically mixed set, however once a required number of vanishing moments has
been reached one can gain more by maximizing sparsity instead of imposing additional
vanishing moments. The required vanishing moments can be incorporated in the parameterization
as discussed in Chapter 5. The designed wavelets are displayed in Figure 6.3.
Once again wavelets strongly resembling the Daubechies 2 wavelets have been found as
an optimum relatively often.
6.1.2 QT time measurement using designed multiwavelets
As explained in Section 2.3 ECG signals can be separated into various segments. The
QRS segment physically corresponds with the depolarization of the ventricles, which
causes the ventricles to contract, and meanwhile atrial repolarization occurs, but due to
the muscle mass of the ventricles the ventricular activity will be dominant. During the

114 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
0.3
0.2
0.1
0
−0.1
−0.2
100 200 300 400 500 600 700 800 900 1000
Figure 6.4: Example of an ECG beat from the PTB database. The beat is extended to
have a length that is a power of two. In order to cope with the end points of the signal
when calculating the wavelet transform, periodic extension is employed.
following ST segment the ventricular muscle cells hold their action potential resulting
in an iso-electric segment. The T wave corresponds to the ventricular repolarization.
The start of the Q wave until the end of the T wave thus corresponds to the total
ventricular activation (depolarization and repolarization) and gives the duration of the
electrical systole. As previously discussed in [70] the QT interval is relevant for a large
number of medical applications. QT prolongation is considered an indicator for sudden
cardiac death, see [99]. The QT interval is used to calculate the beat-to-beat variability
of repolarization (BVR) which currently is a an important indicator for some cardiac
pathologies, see [111]. The QT interval is also linked to non-cardiac pathologies such as
diabetic autonomic neuropathy, see e.g. [41].
The accurate measurement of the QT time is a challenging task and was the subject
of the PhysioNet/Computers in Cardiology Challenge 2006 [88]. Many recent methods
[75, 57, 4] attempt to exploit the information that is contained in multiple leads since it
shows a different projection of the same phenomenon (see Section 2.3).
Using the orthogonal multiwavelet design approach from Section 5.3 it is possible
to construct a multiwavelet such that wavelet function in the multiwavelet structure
is designed for the QRS complex and the other wavelet function in the multiwavelet
structure is designed for the T wave, aiming at distinguishing them in the wavelet domain.
In Figure 6.5 the design of a multiwavelet for the simultaneous detection of both
QRS complexes and T waves is displayed. In the top left of the figure the stationary
wavelet decomposition with the first wavelet in the multiwavelet structure is shown.
The white dashed rectangle is a time-scale mask, used during the optimization, where
the time component corresponds to the QRS complex. In the lower left subfigure the
absolute values of wavelet coefficients of this decomposition are displayed. In the top
right of Figure 6.5 the stationary wavelet decomposition using the second wavelet in the
multiwavelet structure is displayed. The rectangle is the time-scale mask used during
the design, where the time component corresponds to the location of the T wave.
Because a large number of different morphologies can manifest in an ECG signal
(see Section 2.3) it is interesting to see whether a multiwavelet can be designed that is

6.1. APPLICATIONS OF WAVELET DESIGN IN CARDIOLOGY 115
Wavelet coefficients 1 Wavelet coefficients 2
abs(Wavelet coefficients 1) abs(Wavelet coefficients 2)
Figure 6.5: Design of a multiwavelet for the simultaneous detection of the QRS complex
and the T wave for the ECG beat in Figure 6.4. In the upper two figures the detail
coefficients are given at full resolution (SWT) from coarse to fine, and in the lower two
figures the corresponding absolute values. The solid rectangle is a time-scale mask that
indicated the region of interest with respect to the QRS complex. The dashed rectangle
is the mask corresponding to the T wave.
applicable in a number of different situations (morphologies). To this end the criterion
for wavelet design has to be modified a bit. Instead of calculating the multiwavelet
decomposition of a single prototype signal and calculating the l 1 or the l 4 norm of this
decomposition, the l 1 or l 4 norms of the decompositions of a vector of prototype signals
is calculated yielding a vector of criterion values (v 1 , v 2 , . . . , v k ), where k is the number of
prototypes. Next these norms have to be merged into a single design criterion. This can
be accomplished by e.g. taking the sums of the entries in this vector ∑ k
l=1 v k, taking the
maximum over the entries of this vector max(v) or taking the product of the sum and
the maximum of the entries of this vector max(v) · ∑k
l=1 v k. In Figure 6.6 a multiwavelet
that has been designed in this way for a vector of prototypes is displayed. A test signal
was constructed for this figure that stitches a number of different morphologies together.
As a reference the decomposition using the Daubechies 2 wavelet was used. Although
a large number of T waves exhibit improved visibility some of them are still not clearly
visible. This applies in particular to the third T wave in Figure 6.6 which has a low
amplitude.
Since the aim is to detect the Q onset and T end points one could argue that the
wavelet is to be designed to detect these points. For this purpose a new multiwavelet

116 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
D7
D6
D5
D4
D3
D2
D1
D7
D6
D5
D4
D3
D2
D1
D7
D6
D5
D4
D3
D2
D1
(a)
1000 2000 3000 4000 5000 6000 7000 8000
(b)
1000 2000 3000 4000 5000 6000 7000 8000
(c)
1000 2000 3000 4000 5000 6000 7000 8000
(d)
4
2
0
−2
2
0
−2
0.04
0.02
0
−0.02
1.5
1
0.5
1000 2000 3000 4000 5000 6000 7000 8000
Figure 6.6: Design of a multiwavelet for QRS and T wave detection in a range of morphologies.
(a) decomposition with Daubechies 2 wavelet, (b) decomposition with first
multiwavelet, (c) decomposition with second multiwavelet, (d) test signal
was designed, for adapted location of the two masks. This is illustrated in Figure 6.7.
As can be observed in this figure, the detection of the location of the T end point from
this decomposition is not convenient.
For this reason a testing algorithm was developed that performs a multiwavelet transform
on a band-pass filtered ECG signal, considering only a single lead, and then detects
the presence of the QRS complex and the T wave from multiple scales of the stationary
multiwavelet decomposition. The exact location of the QRS onset and T end points is
then determined by a rule-based system that involves the derivative of the single lead
ECG signal. Preliminary results of the current algorithm show that it can detect the location
of the Q onset of 548 beats from the PTB database [16] with a standard deviation
of the error of σ = 15.22ms, for the Q onset point, σ = 30.29ms for the T end point and
σ = 29.87ms for the QT time, with respect to the used gold standard [28] which consists
of manual annotations for beats in the PTB database as published in [25]. It is also
found that the means of the errors are quite high (−2.44ms, −22.82ms and −20.37ms)
respectively and indicate a systematic error that can be corrected at the end of the algorithm
and as a result is not a good measure of the performance of the algorithm. The

6.1. APPLICATIONS OF WAVELET DESIGN IN CARDIOLOGY 117
D7
D6
D5
D4
D3
D2
D1
D7
D6
D5
D4
D3
D2
D1
D7
D6
D5
D4
D3
D2
D1
(a)
1000 2000 3000 4000 5000 6000 7000 8000
(b)
1000 2000 3000 4000 5000 6000 7000 8000
(c)
1000 2000 3000 4000 5000 6000 7000 8000
(d)
1.5
1
0.5
1000 2000 3000 4000 5000 6000 7000 8000
Figure 6.7: Design of a multiwavelet for Q onset and T end detection in a range of
morphologies. (a) decomposition with Daubechies 2 wavelet, (b) decomposition with
first multiwavelet, (c) decomposition with second multiwavelet, (d) test signal
obtained result is not spectacular but most reasonable (see e.g. [75, 57, 4]). Note that
when considering a single lead, the choice of the lead may affect the results [82].

118 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
6.2 Bias field removal from magnetic resonance images using wavelet
design
The wavelet design procedures from Chapter 5 can also be used for 2D applications, such
as image processing, by taking the Cartesian product of 1D wavelet transforms as in [79].
As a concrete example, bias field removal from magnetic resonance images is considered
as published in [64].
Magnetic Resonance Imaging (MRI), or Nuclear Magnetic Resonance Imaging (NMRI)
as it was called in the early days, is widely used in the medical practice. It provides a
much better contrast for soft tissue than for example x-ray / Computed Tomography
(CT). The technique uses a magnetic field that is generated by coils to align the magnetization
of hydrogen atoms. The alignment of this magnetization is altered with radio
waves such that a rotating magnetic field is created that is picked up by the scanner.
The technology is still undergoing major improvements, both when it comes to hardware
performance (such as field strength increase to improve the visibility of tissue
anomalies) and software development (e.g. for sophisticated MR image processing).
An example of such a new development is 4D acquisition: a time-lapsed 3D image of a
subject.
6.2.1 Radio Frequency inhomogeneities in magnetic resonance images
An important type of artifact of interest is the Radio Frequency (RF) inhomogeneity
or bias field artifact; see e.g. [7]. This type of artifact affects MR images and may
seriously hamper reliable diagnosis in practice. The artifact is caused by the fact that
the intensity of the machine magnetic field varies, depending on the scan sequence, tissues
being imaged and the type of coil being used. Regarding the coil type, mainly surface
coil images suffer from this type of artifact (whereas body coil images suffer less from
this type of artifact, but contain more noise). The effect of RF homogeneity artifacts
is that the image suffers from a non-uniform illumination. This expresses itself as, for
example, extended luminescence or an image shade variation, but also as white stripes
as illustrated in for example Figure 6.8.
6.2.2 Bias field removal in magnetic resonance images
One possible approach to remove this artifact, is to first image a phantom (a container
with a known substance) and then to use this to determine the specific magnetic field
intensity variations, see [32, 113]. From the phantom image a degradation model is
computed, which is then used to account for a bias field in other MRI images obtained
under comparable circumstances. In [64], which is used as a basis for this section, an
approach was described that is intended to facilitate bias field removal from a corrupted
image without prior knowledge of the specific field variations. A closely related approach
with similar goals has previously been described in [7, 8] using classical Butterworth
filters.

6.2. BIAS FIELD REMOVAL FROM MR IMAGES USING WAVELET DESIGN 119
Figure 6.8: Knee MRI corrupted with bias field
6.2.3 Wavelet design for RF inhomogeneity detection
In the literature the RF inhomogeneity is commonly viewed and treated as multiplicative
noise. As discussed in [7, 45] one can apply homomorphic filtering by taking the logarithm
of each pixel (ln(·)) of the corrupted image (C ′ = ln(C)) first, to make the bias field
additive such that it can more easily be removed. In our case the image values are in the
range 0−255, and since the logarithm of zero is undefined the image values are increased
by 1.
In order to suppress the bias field from MR images in a wavelet-based approach, in
[64] wavelets are designed by extending the approach discussed in [69] from 1D to 2D.
This extension is carried out by taking the Cartesian product of 1D wavelet transforms
as in [79]. The design is performed by computing the 2D wavelet decomposition of a
prototype image and optimizing it according to an application dependent place-scale
mask. Since the bias field in the present application covers the whole image, and it is
a low frequency artifact, a scale mask is used which uniformly takes only the coarsest
scales into account.
As a prototype, the natural logarithm of an artificial bias field F B from BrainWeb
[18, 72, 29] was used: F C = ln(F B ). To optimize the wavelet design criterion, the
2D wavelet transform of F C was calculated over 5 scales in the following way. First,
the 1D wavelet transform was calculated in the horizontal direction of each row of F C
giving arrays a1 and d1. Next the 1D wavelet transforms were calculated in the vertical
directions of a1 and d1, giving arrays aa1 and ad1 for a1, and da1 and dd1 for d1.
This splits the image into an approximation part aa1 and a horizontal ad1, a vertical
da1 and a diagonal dd1 detail part. The procedure is then reapplied iteratively on the

120 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
aa 1 da 1
Image WT on rows a 1 d 1
WT on cols
ad 1 dd 1
WT on rows
Continue 2D wavelet recursion
aa 2 da 2
da 1
WT on cols
a 2
d 2
da 1
ad 2 dd 2
ad 1 dd 1
ad 1 dd 1
Figure 6.9: 2D wavelet transform
approximation coefficient array aa1 only, yielding the arrays aa2, ad2, da2 and dd2. This
procedure is then applied recursively to aa2. This process is illustrated in Figure 6.9.
The depth of this recursion is limited by the size of the image and the length of the filters.
Once the array aa5 is obtained, the l 4 norm of its entries is calculated, providing the
design criterion V from [64] that is to be maximized by tuning the free filter coefficients:
⎛
V (F C ) = max ⎝ ∑ i
∑
j
|aa5 i,j | 4 ⎞
⎠ (6.1)
This procedure assists in creating a wavelet which pushes the energy of the prototype
bias field into the approximation scale. The key idea is to exploit the fact that the bias
field is a slowly varying artifact. This process is illustrated in Figure 6.10. The reader
should note that the scaling of the prototype bias field is not necessarily the same as the
bias field in the actual image. The actual number of scales in the transformation should
be selected in accordance with the actual resolution of the image. In this manner the
low-pass FIR filter
− 0.0230 + 0.0173z −1 + 0.1533z −2 + 0.1349z −3 + 0.5148z −4 + 0.7898z −5 + 0.0455z −6
− 0.2567z −7 + 0.0165z −8 + 0.0219z −9

6.2. BIAS FIELD REMOVAL FROM MR IMAGES USING WAVELET DESIGN 121
Prototype image
wavelet coefficients
|wavelet coefficients|
wavelet and scaling function
Figure 6.10: Wavelet designed to maximize the l 4 -norm of the approximation coefficients
on scale 5. As a prototype the logarithm of an artificial bias field from BrainWeb was
taken. For the optimization n − 1 = 4 free parameters were used, which provided a FIR
filter of length 2n = 10.
and the associated power complementary high-pass FIR filter
0.0219 − 0.0165z −1 − 0.2567z −2 − 0.0455z −3 + 0.7898z −4 − 0.5148z −5 + 0.1349z −6
− 0.1533z −7 + 0.0173z −8 + 0.0230z −9
were obtained.
6.2.4 Filtering MR images with designed wavelets
In order to filter the actual image, the SWT is applied. This technique that is discussed
in Section 3.3 is extended to two dimensions by means of a Cartesian product. The
advantage of this approach is that artifacts that can appear with the regular wavelet
transform (see e.g. Figure 6.11) are avoided when reconstructing the bias field from the
approximation coefficients only. A discussion on artifacts related to such smoothness
issues can be found in, e.g., [107, Chapter 10].

122 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
0 1 2
3 4 5
Figure 6.11: Original image and reconstructed images with each time the n finest detail
coefficients omitted where n is the title of the image. Note that artifacts are introduced,
due to a lack of smoothness.
Original image C
ln(C+1)
50
50
100
100
150
150
200
200
250
50 100 150 200 250
250
50 100 150 200 250
SWT Approx. coefs.
Wavelet decomposition of ln(C+1)
50
50
100
100
150
150
200
200
250
50 100 150 200 250
250
50 100 150 200 250
Figure 6.12: In clockwise order: the MRI image of a hip, the logarithm of this image, a
5-scale wavelet decomposition of image, and the approximation coefficients of the SWT
of the image.

6.2. BIAS FIELD REMOVAL FROM MR IMAGES USING WAVELET DESIGN 123
Another technicality which requires attention is that when applying homomorphic
filtering, streak artifacts are created at the boundary between tissue and background.
Assume that there is a mask M that indicates the tissue region. In order to avoid these
artifacts normalized convolution [45] is used. The natural logarithm ln(F ) of the bias field
F in the original image O in Figure 6.12 can be reconstructed from the approximation
coefficients of the stationary wavelet decomposition of ln(O)•M, where the • denotes the
Hadamard product, i.e., the entry-wise product. The mask M is filtered in exactly the
same manner giving w(M), where w(·) indicates that it has been filtered using a wavelet
approach. The actual estimated bias field is then obtained by first applying normalized
convolution followed by taking the exponential of each entry exp(·):
( )
w(ln(O) • M)
F = exp
. (6.2)
w(M)
The obtained bias field is shown in the center of Figure 6.13. The original image at the
left side of Figure 6.13 is divided by this bias field to obtain the restored image at the right
side of Figure 6.13. For the criteria discussed in [64] the image needs to be segmented.
For each segment i the mean of the pixel values in the segment µ i are calculated as well as
the standard deviation σ i . Next as previously discussed in [7] the coefficient of variance
cv i = σi
µ i
of each segment and the coefficient of contrast cc i,j = µi
µ j
between each adjacent
pair of segments is calculated. The measures of performance used here are, as discussed
in [64], the minimum of the absolute values of the coefficients of contrast of the adjacent
segments min(bcc|bcc > 0) = min segements i,j adjacent {abs(ln(cc i,j ))}, and the mean of
the negative logarithm of the coefficients of variance mean(lcv) = mean i {− ln(cv i )}. An
overall criterion can be computed as opc = min(bcc|bcc > 0)mean(lcv). In terms of these
performance measures a score of min(bcc|bcc > 0) = 0.392 was obtained for the original
image and min(bcc|bcc > 0) = 0.588 for the reconstructed image. For the reduction in
variance the original image had a score of mean(lcv) = 0.0695 and the reconstructed
image a score of mean(lcv) = 1.024. The respective products are opc = 0.0272 and
opc = 0.602, which constitutes a major improvement for the reconstructed image.
The current approach appears to work well on images with strong small details. The
results for e.g. brain MR images are currently not as good. If the wavelet transform
for these images is calculated on more scales one can see an improvement in the results.
However the maximum scale that can be calculated is limited by both the image size
and the filter size. The relevant components of the image obviously live at the same
coarse scale as the bias field. A possible way to overcome this problem is to design
multiwavelets (Section 5.3) and to simultaneously optimize one wavelet for the bias field
and another orthogonal wavelet for the relevant image such that they become separable.
This approach is more flexible than the current one.

124 CHAPTER 6. BIOMEDICAL APPLICATIONS OF WAVELET DESIGN
Original Bias Restored Image
Figure 6.13: From left to right: the original image, the bias field as detected by the
method and the filtered image after removal of the multiplicative bias field.

Chapter 7
Conclusions and directions for further
research
Technological advancements in the field of biomedical engineering help to lengthen the
lifespan of patients that suffer from cardiovascular related illnesses. One of such advancements
are implantable devices such as pacemakers. Battery life is a critical issue
for bio-implantables. To reduce the power demand the frequency of therapies need to
be reduced, i.e., good sense amplifiers are crucial and the power consumption of these
sense amplifiers needs to be as low as possible. In recent years the technique of wavelet
transforms that offer simultaneous time-frequency resolution with a zoom-in property,
that offer a choice of basis and that can be used to measure the regularity of signals, has
gained popularity in the field of biomedical engineering. Wavelets are a signal processing
technique that can make sense amplifiers in pacemakers more powerful. However,
A/D conversion is power consuming and for low-power applications a low resolution A/D
converter is required. This implies that one should perform as many computations as
possible in the analog domain. It is possible to approximate the wavelet transform by
matching the impulse response of a linear system with the time-reversed, time-shifted
wavelet function. Previously this matching was established by means of Padé approximation.
It is discussed however that this technique has a number of drawbacks:
• The matching is performed in the Laplace domain in the neighborhood of a point
s 0 which has to be chosen in advance.
• Stability of the filter is not automatically guaranteed. The choice s 0 = 0 will help
to obtain stability but unfortunately it is likely to give a poor fit at the beginning
of the impulse response in the region where the wavelet “lives”.
• The choice of the degrees of the numerator and denominator polynomials which
constitute the Padé approximation is not a trivial problem and may strongly influence
the quality of the results.
125

126 CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FURTHER RESEARCH
A number of variations to Padé approximation are discussed that attempt to cope with
these problems, but the drawback remains that the quality of the approximation is not
measured directly in the time domain, but in the Laplace domain, with the consequence
that it does not allow for a direct interpretation in system theoretic terms.
In this thesis a novel approach is introduced based on L 2 approximation. This approach
offers a number of conceptual and practical advantages:
• The wavelet transform involves the L 2 -inner product between the wavelet function
and the signal with an arbitrary time shift. The L 2 approximation approach treats
all time instances equal and is appropriate for reliably computing L 2 -inner products.
It allows one to determine an error bound for the approximations obtained in this
way. Optionally it allows for the use of weighing.
• Due to Parsevals’s equality the L 2 -norm allows for a description in both the time
domain and in the Laplace domain, which we use to our advantage.
In order to approximate wavelet functions with the impulse response of linear systems
a parameterization of these impulse responses was proposed that is appropriate for the
approximation of wavelet functions. Additionally it is shown that it is required to enforce
a first vanishing moment in order to avoid a bias in the approximate wavelet transform.
It is shown how the condition for this vanishing moment translates in terms of the
parameters that are used to build linear systems which effectively implement a single
scale of the wavelet transform.
Since the optimization surface corresponding to the wavelet approximation error may
contain multiple local optima, a procedure is proposed to deterministically find a good
starting point for the iterative local search optimization routine. The L 2 approximation
methodology was demonstrated on the Gaussian wavelet, the Morlet wavelet, the Mexican
hat wavelet, the Daubechies 3 wavelet, the Daubechies 7 wavelet and the Coiflet 5
wavelet, of which only one has been successfully implemented directly using the Padé
approach. The results of these approximations are quite satisfactory as a specified approximation
quality is now obtained with approximations of lower order than previously
found with Paé approximation. However it may still happen that the approximation corresponds
to an unsatisfactory local optimum. One may approach this problem in several
ways:
• To develop a different model order reduction technique that specifically attempts
to preserve the shape of the impulse response, rather than to optimize one of the
currently available criteria for model order reduction.
• To employ or develop a global optimization technique for the problem at hand.
General global optimization techniques, such as simulated annealing, may have
the drawback of being computationally intensive. For L 2 approximation, however,
theoretical results are available in discrete-time which guarantee the existence of
only a finite number of local optima.

127
• To develop a practical approach which employs a large number of different starting
points which are well distributed over the class of candidate approximations.
The choice of a specific wavelet may have substantial impact on the performance of
signal processing algorithms such as detection an compression algorithms. This raises
the research question of how to select an appropriate wavelet for the signals and problem
at hand. In order to support this decision it was argued that for applications such as
detection and compression, a sparse representation in the wavelet domain is beneficial.
To obtain sparsity we have persued the principle of maximization of the variance of
the wavelet coefficients. For orthogonal wavelets from filter banks using finite-length
signals, the energy of the detail and approximation coefficients, computed with periodic
extension, will be a constant equalling the energy of the signal. As a result it is possible
to either maximize the variance of the absolute values of the wavelet coefficients which
is shown to boil down to minimizing their l 1 -norm, or to maximize the variance of the
squared wavelet coefficients which comes down to maximization of their l 4 -norm. These
criteria can be used as optimization criteria for the design of wavelets.
As a parameterization of orthogonal wavelets, the existing lattice structure of wavelet
filters in polyphase form was used. Additional constraints to enforce vanishing moments
are worked out and made explicit in terms of the parameterization used. The first vanishing
moment can be built-in by eliminating a parameter. A second vanishing moment
is given in terms of a constraint on the parameters. This novel combination of the
parameterization, the newly introduced design goals and the enforcement of vanishing
moments allows for the design of orthogonal wavelets. It was shown that there exist
local optima on the optimization surface and appropriate measures need to be taken to
arrive at satisfactory results. The applicability of the approach was demonstrated two
examples in Chapter 6: on the detection of the QRS complex in ECG signals and on
a 2D application: bias field removal in MR images. For low-order wavelet filters, the
property to have many vanishing moments can be beneficial in a morphologically mixed
set of signals. However this does not hold in general and once a limited number of vanishing
moments has been reached one can gain more by maximizing sparsity instead of
imposing additional vanishing moments.
One of the major achievements of this thesis is that we have developed a complete
procedure to go from a prototype signal, related to a specific application, to a linear filter
which performs approximate wavelet analysis in the analog domain, ready for implementation
in low-power hardware:
• Wavelet filter design by optimization with a sparsity criterion over a parameterized
class
• Computation of the wavelet and scaling function
• L 2 approximation to construct a linear filter
When one wants to distinguish between morphologies in a given signal, multiwavelets
can be employed. Orthogonal multiwavelets can, as previously discussed in the literature

128 CHAPTER 7. CONCLUSIONS AND DIRECTIONS FOR FURTHER RESEARCH
and further explained in this work, be formulated in polyphase as lossless systems. In
this work a method is introduced that allows for the recursive construction of orthogonal
multiwavelets by employing balanced realizations of lossless systems and the reversed
tangential Schur algorithm. At the heart of the tangential Schur algorithm are linear
fractional transforms that build a a lossless polyphase matrix in state-space form by
means of a recursive procedure. This provides us with a general framework to deal with
orthogonal multiwavelets which encompasses all the existing parameterizations in the
literature, and which has a number of advantages from a computational point of view.
Numerical experience shows shows that a well behaved implementation is feasible, as
orthogonal matrices are employed.
In this work it is also presented how balanced vanishing moments can be built in
to obtain a valid multiwavelet structure that can be used on arbitrary signals without
the need of prefiltering. The first balanced vanishing moment corresponds to an interpolation
condition on the unit circle. Starting from a low-order system with a built-in
balanced vanishing moment, it is shown how the tangential Schur algorithm can be used
to recursively build a lossless system of the desired order, which then gives rise to a multiwavelet
structure. Effectively these ingredients provide a parameterization of orthogonal
multiwavelets with compact support and built-in balanced vanishing moments for which
no parameterization was previously available in the literature. This facilitates the design
of orthogonal multiwavelets. Additionally, a design procedure is developed which
involves masks to highlight different morphologies in a signal to which multiwavelets are
adapted.The potential of this approach is demonstrated on the detection of the Q onset
and T end points in ECG signals for the purpose of QT interval estimation. For image
processing the multiwavelets have been found to be currently too irregular. Additional
balanced vanishing moments are known to enhance the regularity of multiwavelets. In
the future additional interpolation conditions on the unit circle might be imposed in
order to enforce such additional balanced vanishing moments.

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Summary
Each year a large number of people decease due to cardiovascular disorders. Due to
medical advancements the contribution of these disorders to the total mortality is reduced.
One of these advancements is a joint medical and engineering development: the
implantable devices, among which the pacemaker. Early pacemakers electro-stimulated
the heart at a fixed rate, ensuring that the heart contracts according to that rate and
pumps blood throughout the body. This approach however has a number of drawbacks.
First of all, the continuous stimulation of the heart is power consuming, which is a major
issue for a device that is surgically inserted into the body and is not easily rechargeable.
Secondly, unnecessary stimulation of the heart can be harmful. Thirdly, the fixed rate of
the pacemaker can interfere with the intrinsic rate of the heart. Nowadays pacemakers
are equipped with a sensing circuit that monitors the electrical currents on the heart.
To accomplish this a monitoring mechanism is required that performs signal processing,
leading to a decision step, in which the decision is made whether or not to stimulate the
heart. Considering that this monitoring circuit is always active, it must be ensured that
this circuit is not power-consuming. Obviously this circuit must also be robust since a
patient’s life depends on it.
A relatively young signal processing technique that is interesting for the use in pacemakers
is the so-called “wavelet transform”. With this technique a signal can simultaneously
be represent in both the time- and frequency-domain. It is easy to implement
in a computer. However, a computer generally operates in the digital domain, whereas
the sensor information is in the analog domain. An analog signal thus has to be converted
to the digital domain, which is a power-consuming operation. An energy saving
solution is to implement the wavelets in the analog domain, and in this manner to reduce
the amount of analog to digital conversion. To achieve this, wavelets have to be
approximated by means of linear systems that can be applied in microelectronics. This
is not a trivial task. In the current work it is discussed why L 2 approximation is a
relevant technique. A complete approach is discussed to approximate wavelet functions
(associated with both continuous and discrete-time wavelets) with this technique. Since
the optimization surface may contain various local optima, it is discussed how a good
starting point for a local search algorithm can be found. Various examples illustrate the
139

140 SUMMARY
flexibility of this approach.
The simultaneous time-frequency representation characteristic is not the only distinction
between wavelets and Fourier transforms. Unlike Fourier transforms, wavelet
transforms offer a choice of bases. Two criteria, that operate in the wavelet domain, are
introduced to determine how good a certain wavelet is for compressing or detecting a
given signal. These criteria are then used, along with a parameterization of orthogonal
wavelets based on “polyphase filters” and the ‘lattice structure”, to design custom
wavelets for an application at hand. Not only scalar wavelets are of interest, but also
multiwavelets. These involve multiple orthogonal wavelet and scaling functions that enable
them to separate orthogonal components in a signal. A parameterization in terms of
“lossless systems” is introduced for these multiwavelets. This parameterization is more
general than the parameterizations that are known from the literature. For a number
of these parameterizations it is discussed how these follow from the introduced parameterization
as special cases. For the new parameterization it is additionally discussed
haw balanced vanishing moments can be built-in, which is required in order to use these
designed multiwavelets directly on measured signals. These balanced vanishing moments
were not explicitly build into earlier parameterizations.
To demonstrate the potential of the discussed techniques, three examples are worked
out. Firstly the designed scalar wavelets are used to detect the QRS complex in an ECG.
Experiments show that the designed wavelets indeed offer advantages. An interesting
observation is that the Daubechies 2 wavelets is often found as an optimum. As a second
application designed multiwavelets are used to simultaneously distinguish the Q and the
T peak in ECGs. A rule-based decision algorithm that has been designed as a proof
of principle readily shows promising results. As a third application wavelet design is
used to facilitate the processing of MR images. Using wavelet filtering, low-frequency
multiplicative noise is successfully removed from images of the pelvis and the knee. The
current technique is not as successful on MR images of the brain. Recommendations to
increase the performance are made.

Samenvatting
Jaarlijks sterft een groot aantal mensen ten gevolge van cardiovasculaire aandoeningen.
Door vooruitgang op het gebied van de medische wetenschap, waaronder meer specifiek
implanteerbare apparaten, wordt het aandeel in de totale sterfte verkleind. Een van de
eerste van dergelijke implanteerbare apparaten is de pacemaker. De vroege uitvoeringen
geven met een vast ritme elektrische pulsen aan het hart, waardoor dit met het vastgestelde
ritme samenknijpt en bloed door het lichaam pompt. Hier kleeft echter een
aantal nadelen aan. Zo kost het voortdurend stimuleren van het hart veel energie, wat
een groot probleem vormt omdat een dergelijk apparaat chirurgisch in het lichaam van
de patiënt is geplaatst en daardoor moeilijk opgeladen kan worden. Ook kan het onnodig
stimuleren van het hart schadelijk zijn. Daarnaast kan het vaste ritme van de pacemaker
interfereren met de frequentie waarop het hart zelf wil samentrekken. Tegenwoordig
zijn pacemakers daarom uitgerust met sensoren die de elektrische stromen op het hart
registreren die vervolgens worden gebruikt in een beslissingsstap waarin wordt besloten
of het hart gestimuleerd moet worden of niet. Dit vereist een monitoringsmechanisme
dat signaalverwerkingstaken uitvoert. Aangezien het monitoring circuit altijd actief is,
is het belangrijk dat dit zuinig met de energie omgaat. Daarnaast moet het de juiste
beslissingen nemen, omdat deze van levensbelang kunnen zijn.
Een relatief recente signaalverwerkingstechniek die interessant is voor het gebruik in
pacemakers is de zogenaamde "wavelet transformatie". Deze techniek maakt het mogelijk
om een signaal tegelijkertijd in tijd en frequentie af te beelden. Het is gemakkelijk om
deze techniek in een computer te implementeren. Echter een computer werkt digitaal en
de sensor informatie in pacemakers is analoog. Er moet dus een analoog signaal omgezet
worden in een digitaal signaal wat veel energie vergt. Een energiezuinige oplossing is om
de wavelets op een analoge manier te implementeren en zo de analoog/digitaal omzetting
te beperken. Om dit te doen moeten de wavelet functies benaderd worden met behulp
van lineaire systemen die in de microelektronica toepasbaar zijn. Dit is echter geen
triviale opgave. In het huidige werk wordt beargumenteerd waarom L 2 approximatie
een goede aanpak is hiervoor. Een complete aanpak om wavelet functies (zowel van
continue als discrete wavelets) met deze techniek te benaderen is uitgewerkt. Aangezien
het optimalisatieoppervlak diverse locale optima kan bevatten, wordt er besproken hoe
141

142 SAMENVATTING
een goed startpunt voor een locale zoektechniek kan worden gevonden. Met diverse
voorbeelden wordt de flexibiliteit van de aanpak aangetoond.
Buiten het feit dat wavelets een signaal tegelijkertijd in termen van tijd als frequentie
laten zien is er nog een ander duidelijk verschil met een klassieke techniek als Fourier
transformaties: er zijn geen vaste basissen en er is keuzevrijheid. Dit geeft meteen een
keuzeprobleem. Er worden twee criteria behandeld die in het wavelet domein bepalen
hoe goed een gegeven wavelet is om een bepaald signaal te comprimeren of te detecteren.
Daarnaast worden deze criteria met een parameterizatie op basis van “polyphase filters”
en de “lattice” structuur van discrete, orthogonale wavelets, gebruikt om wavelets te
ontwerpen. Naast reguliere wavelets zijn ook zogenaamde “multiwavelets” interessant.
Deze beschikken over meerdere orthogonale wavelet en schalingsfuncties en kunnen diverse
orthogonale componenten in een signaal onderscheiden. Voor deze multiwavelets
is een parameterisatie opgezet in termen van zogenaamde “lossless” systemen. Deze parameterisatie
is algemener dan eerdere parameterisaties die bekend zijn uit de literatuur.
Van een aantal van deze bestaande parameterisaties wordt besproken hoe ze volgen uit
de geïntroduceerde, algemenere parameterisatie. Daarnaast wordt besproken hoe met de
geïntroduceerde parameterisatie gebalanceerde momenten kunnen worden ingebouwd,
hetgeen noodzakelijk is om de, met deze parameterisatie, ontworpen multiwavelets direct
op gemeten signalen toe te passen. Deze gebalanceerde momenten werden in eerdere
parameterisaties niet ingebouwd.
Om de kracht van de besproken technieken te demonstreren wordt een drietal voorbeelden
uitgewerkt. Als eerste worden ontworpen wavelets gebruikt om het QRS complex
in een hartsignaal te detecteren. Uit numerieke gegevens blijkt dat deze ontworpen
wavelets inderdaad voordelen bieden. Een opmerkelijk resultaat is dat een bepaalde
wavelet, nl. de Daubechies 2 wavelet dikwijls als optimum gevonden wordt. Als tweede
applicatie worden ontworpen multiwavelets gebruikt om tegelijkertijd de Q piek en de T
piek in een hartsignaal te onderscheiden. Een regelgebaseerd detectiealgoritme dat als
voorbeeld is ontworpen laat reeds bemoedigende resultaten zien. Als derde toepassing
worden wavelets ontworpen en gebruikt voor het verwerken van afbeeldingen afkomstig
van een MRI scanner. Met behulp van deze technieken wordt succesvol laagfrequente
multiplicatieve ruis verwijderd bij afbeeldingen van het bekken en de knie. Op MRI
afbeeldingen van hersenen is de huidige techniek minder succesvol, echter aanbevelingen
om dit verder te verbeteren worden gedaan.

Curriculum Vitae
1978 Born on 10 April 1978 in Maastricht, The Netherlands
1990–1997 Secondary School, Jeanne d’Arc College Maastricht, HAVO/VWO
1997–2001 Master of Science in Knowledge Engineering, business mathematics
major. Joint programme of the Maastricht University (The Netherlands)
and Limburgs Universitair Centrum (Nowadays Hasselt University,
Belgium). Obtained the Belgian diploma “Kandidaat Informatica”
(Computer Science) from the Limburgs Universitair Centrum
in 1999. Attended a special four-week programme on knowledge
engineering and computer science in 1999 at Baylor University,
Waco, Texas. Master thesis at Medtronic Bakken Research Center
b.v., Maastricht, The Netherlands, on the mathematical modeling of
atrial fibrillation, completing a five years dual track at the Maastricht
University in four years in 2001.
2001–2004 Joint position at MaTeUM b.v., Maastricht, The Netherlands as a developer
and at the Maastricht University, Faculty of General Sciences
for research on simulation.
2004–2008 Ph.D. Student on the STW funded BioSens project at the Maastricht
University, Faculty of Humanities and Sciences, MICC, Department
of Mathematics.
2007–2008 Half-time position as a lecturer at the Maastricht University, Faculty
of Humanities and Sciences, MICC, Department of Mathematics.
2008–current Assistant Professor at the Maastricht University, Faculty of Humanities
and Sciences, Department of Knowledge Engineering.
143

List of Symbols
and Abbreviations
Symbol Description Definition
• Hadamard product page 123
∗ convolution operator page 17
†
Hermitian transpose page 42
↓2 downsampling by 2; (↓2x) k = x 2k page 28
A system matrix for a linear system in state-space representation
page 21
a k approximation or scaling coefficients page 31
B input matrix for a linear system in state-space representation
page 21
b k detail or wavelet coefficients page 31
C output matrix for a linear system in state-space representation
page 21
c k scaling filter coefficients page 30
δ[n] Kronecker delta page 18
δ(t) Dirac delta page 18
D direct feed-through matrix for a linear system in statespace
page 21
representation
d k wavelet filter coefficients page 30
e n n th standard basis vector page 99
exp element-wise exponential page 123
F U,V mapping for a proper rational matrix page 92
H(E) Halmos extension page 95
H Hankel matrix page 61
h(t) impulse response page 20
H(s) transfer function page 20
H 0(z) wavelet low-pass filter page 28
H 1(z) wavelet high-pass filter page 28
H p(z) polyphase matrix in z page 35
145

146 LISTS OF SYMBOLS AND ABBREVIATIONS
Symbol Description Definition
i complex number page 16
1 A(x) indicator function page 54
L Laplace transform page 18
ln Element-wise natural logarithm page 119
ω frequency variable page 16
φ phase angle page 16
u(t) input for a system page 19
ϕ(t) scaling function page 31
ϕ(t) multi scaling function page 40
ψ(t) wavelet function page 26
˜ψ(t) time-reversed and time-shifted wavelet function page 47
ˇψ(t) causal time-reversed and time-shifted wavelet function page 47
ψ(t) multi wavelet function page 40
σ scale page 26
T linear fractional transformation page 93
W (τ, σ) wavelet transform page 26
x(t) state vector page 21
y(t) output for a system page 19
Y (iω) Laplace transform of y(t) for s = iω page 18
Y (ω) Fourier transform of y(t) page 18
Y (s) Laplace transform of y(t) page 18
Z Z-transform page 19
Abbreviation Description Definition
3D three-dimensional page 10
A/D analog-to-digital page 15
AF atrial fibrillation page 14
AV atrio-ventricular page 9
BVR beat-to-beat variability of repolarization page 114
CT computed tomography page 118
CWT continuous wavelet transform page 26
DC direct current page 21
DTL dynamic translinear page 43
DWT discrete wavelet transform page 27
ECG electrocardiogram page 10
EKG electrocardiogram page 10
FIR finite impulse response page 23
Hz Hertz page 16
IECG intracardiac electrocardiogram page 10
LA left atrium page 8
LFT linear fractional transformations page 93
LTI linear time-invariant page 19
LV left ventricle page 8

147
Abbreviation Description Definition
MA moving average page 23
MIMO multi-input multi-output page 21
MIT-BIH Massachusetts Institute of Technology - Beth Israel page 14
Deaconess Medical Center
MR magnetic resonance page 118
MRA multi-resolution analysis page 31
MRI magnetic resonance imaging page 118
NMRI nuclear magnetic resonance imaging page 118
pdf probability density function page 27
RA right atrium page 8
RF radio frequency page 118
RV right ventricle page 8
SA sino-atrial page 9
SISO single-input single-output page 20
VF ventricular fibrillation page 14
ZOH zero-order-hold page 18

Index
l 1 -norm, 73
l 4 -norm, 73
A/D converter, 15, 43
action potential, 9
admissability condition, 26
aliasing, 28
all-pass systems, 41, 90
alternating flip, 30
alternating signs, 29
analog, 15
analysis, 16
angular frequency, 16
anisotropy, 10
approximation
L 2 , 51
coefficients, 28
error, 34
Padé, 48
approximation space, 31
arrhythmia, 10
asymptotic stability, 20, 91
atrial fibrillation, 14
atrio-ventricular node, 9
atrium, 8
balance and truncate, 62
balancing, 60, 98, 101, 108
bandlimited, 16
bandwidth, 16
bias field artifact, 118
Bode plot, 16
Bohl functions, 53
canonical form, 21
cardiovascular, 7
cascade of filter banks, 31, 75
Cauchy-Schwarz inequality, 52
causality, 20
closure, 31
coefficient of contrast, 123
coefficient of variance, 123
coefficients
approximation, 28
scaling, 28
wavelet, 28
compact support, 29, 40
Computed Tomography, 118
conjugate transpose, 42
conservation of energy, 29, 72
continuous
Fourier transform, 16
time, 15
wavelet transform, 26
controllability Grammian, 60
controllability matrix, 61
controllable companion form, 61
convolution, 17, 45
convolution kernel, 51
critical sampling, 28, 36
Daubechies wavelets, 34
decision stage, 8
decomposition, 31
defibrillator, 14
depolarization, 8
depolarized, 9
design, wavelet, 72
detail
coefficients, 28
signal, 33
detail space, 32
digital, 15
dilation, 26
equation, 32
Dirac delta, 18–20, 59
direct feedthrough, 21
discrete
149

150 INDEX
Fourier series, 17
Fourier transform, 17
time, 15
wavelet transform, 27
domain, 15
double-shift orthogonality, 30
downsampling, 28
dyadic points, 32
dyadic scales, 31
dynamic translinear circuits, 43
dynamical matrix, 21
ECG, 10
intracardiac, 10
subcutaneous, 10
surface, 10
electrical systole, 13
electrocardiogram, 10
elementary J-inner factors, 93
energy distribution, 73
Euler’s formula, 16
even phase, 35
exponent
Hölder, 34
Lipschitz, 34
fibrillation
atrial, 14
ventricular, 14
filter bank, 27, 72
filter synthesis, 48
final value theorem, 19
finite energy, 16
finite impulse response, 23, 29, 89
Fourier
series, 16
discrete, 17
transform, 15
continuous, 16
discrete, 17
windowed, 18
frequency domain, 15
frequency response function, 22
full rate, 34
function, vector, 40
Gabor transform, 18
Gaussian wavelet, 27
global optimization, 51, 57, 86
Hölder exponent, 34
Haar wavelet, 28
Hadamard product, 123
half rate, 35
Halmos extension, 95
Hankel matrix, 61
Hankel singular values, 60
heart, 8
Heisenberg uncertainty rectangle, 26
Hermitian transpose, 42
high-pass filter, 28
homomorphic filtering, 119
implantable devices, 7, 43
impulse
continuous-time, 18, 20
discrete-time, 18
response function, 20
impulse response, 45
indicator function, 54
initial phase angles, 16
initial value theorem, 18
inner-product, 51
input matrix, 21
input vector, 21
integral wavelet transform, 26
intracardiac ECG, 10
iteration scheme, 32
Kronecker delta, 18, 20, 59
L 2 -approximation, 51
Laplace transform, 18
lattice form, 75
lead, 10
linear fractional transform, 93
linear phase, 22, 34
linear system, 19
time-invariant, 19
Lipschitz exponent, 34
local optima, 82
lossless systems, 41, 89, 90
low-pass filter, 28
Lyapunov-Stein equations, 60
magnetic resonance imaging, 118
Mallat’s algorithm, 31
matrix exponential, 53
McMillan degree, 20
Mexican Hat wavelet, 27
modulation matrix, 29
mother wavelet, 34
moving average filters, 23
multi-resolution, 25, 31
multiplicative noise, 119
multiwavelet, 34, 40, 89
design, 89
parameterization, 42

INDEX 151
myocardium, 8
normalized convolution, 123
nuclear magnetic resonance imaging, 118
Nyquist frequency, 15
Nyquist rate, 15
observability grammian, 61
observability matrix, 61
odd phase, 35
orthogonal complement, 32
orthogonal filter banks, 29
orthogonal wavelets, 75
orthogonality condition, 29
orthonormal wavelets, 25
output matrix, 21
output vector, 21
overcomplete wavelet transform, 37
P wave, 12
pacemaker, 8
pacemaker cells, 9
Padé approximation, 45
Padé approximation, 48
Parseval’s identity, 51
perfect reconstruction, 29
phantom, 118
phase, 34
polarized, 8
poles, 20
polyphase filters, 34, 75, 76
polyphase matrix, 35
power, 16
power complementary, 29
PP interval, 13
PQ segment, 12
proper rational transfer function, 20
pseudofrequency, 26
QRS complex, 12
QT interval, 13, 114
quadrature mirror filters, 30
radio frequency inhomogeneity, 118
reduced rate, 35
region of convergence, 18
regularity, 34
repolarization, 9, 12
resting potential, 8
rhythm, 14
rotation matrix, 76
RR interval, 13
sampling
frequency, 15
rate, 15
scaling
coefficients, 28
filter coefficients, 30
function, 31
Schur
form, 98
tangential algorithm, 93
vector, 94
segments (ECG), 12
separation of exponentials, 54
short time Fourier transform, 18
signal processing, 7
similarity transformation, 21
sino-atrial node, 9
sinusoidal fidelity, 22
Smith-Barnwell orthogonality conditions, 40
space, approximation, 31
space,detail, 32
sparsity, 72
ST segment, 12
stability, 50, 54, 90
starting point, 57
state matrix, 21
state vector, 21
state-space, 20
dimension, 21
representation, 21, 54
stationary wavelet transform, 37
stationary wavelet transform, polyphase, 38
step response function, 20
strictly proper rational function, 45
subcutaneous ECG, 10
sudden cardiac death, 14
surface ECG, 10
syncytium, 14
synthesis, 16
system matrix, 21
T wave, 12, 114
tangential Schur algorithm, 93
Taylor series expansion, 48
time domain, 15
time-frequency localization, 25
time-invariant systems, 19
time-shifting, 26, 47
transfer function, 20
proper rational, 20
transform
Fourier, 15
continuous, 16
discrete, 17

152 INDEX
windowed, 18
Gabor, 18
Laplace, 18
wavelet, 25
continuous, 26
discrete, 27
truncation error, 47
uncertainty rectangle, 18, 26
upsampling, 28
vanishing moments, 33, 55, 79, 80, 101
variance maximization, 73
vector function, 40
vectorfunctions, 40
ventricle, 8
ventricular fibrillation, 14
wavelet, 71
approximation, 45
basis, 33, 71
choosing a, 72
coefficients, 28
design, 72
equation, 32
filter coefficients, 30
function, 31, 32
Gaussian, 27
Mexican Hat, 27
multi, 40, 89
orthonormal, 25, 75
transform, 25
continuous, 26
discrete, 27
overcomplete, 37
stationary, 37
wavelets
Daubechies, 34
waves (ECG), 12
windowed Fourier transform, 18
zero-order hold, 18, 59
zeros, 20
zoom-in property, 26